mutual information, fisher information, and population coding1732 nicolas brunel and jean-pierre...

LETTER Communicated by Alexandre Pouget

Mutual Information Fisher Information and PopulationCoding

Nicolas BrunelJean-Pierre NadalLaboratoire de Physique Statistique de IrsquoENSdagger Ecole Normale Sup Acircerieure 75231Paris Cedex 05 France

In the context of parameter estimation and model selection it is only quiterecentlythat a direct link between the Fisher information and information-theoretic quantities has been exhibited We give an interpretation of thislink within the standard framework of information theory We show thatin the context of population coding the mutual information between theactivity of a large array of neurons and a stimulus to which the neuronsare tuned is naturally related to the Fisher information In the light of thisresult we consider the optimization of the tuning curves parameters inthe case of neurons responding to a stimulus represented by an angularvariable

1 Introduction

A natural framework to study how neurons communicate or transmit in-formation in the nervous system is information theory (see eg Blahut1988 Cover amp Thomas 1991) In recent years the use of information theoryin neuroscience has motivated a large amount of work (eg Laughlin 1981Linsker 1988 Barlow Kaushal amp Mitchison 1989 Bialek Rieke de Ruytervan Steveninck amp Warland 1991 Van Hateren 1992 Atick 1992 Nadalamp Parga 1994) A neurophysiologist often asks in an informal sense howmuch information the spike train of a single neuron or of a population ofneurons provides about an external stimulus For example a high activityof a CA3 hippocampal neuron may tell with good precision where a ratis in an environment Information theory provides mathematical tools formeasuring this ldquoinformationrdquo or ldquoselectivityrdquo signals are characterized bya probability distribution and the spike train of a neuron or of a popula-tion is characterized by a probability distribution conditioned by the signalThe mutual information between the signal and the neural representationis then a measure of the statistical dependency between the signal and thespike train(s)

dagger Laboratory associated with CNRS ENS and universities Paris 6 and Paris 7

Neural Computation 10 1731ndash1757 (1998) cdeg 1998 Massachusetts Institute of Technology

1732 Nicolas Brunel and Jean-Pierre Nadal

A related domain which also belongs to information theory is the eldof statistical parameter estimation Here one typically has a sample of ob-servations drawn from a distribution that depends on a parameter or a setof parameters that one wishes to estimate The Cramer-Rao inequality thentells us that the mean squared error of any unbiased estimator of the under-lying parameter(s) is lower bounded by the inverse of a quantity which isdened as the Fisher information (Blahut 1988) This means that the Fisherinformation is a measure of how well one can estimate a parameter froman observation with a given probability law Thus in this sense it is also anldquoinformationrdquo quantity

In spite of the similar intuitive meanings of these two quantities an ex-plicit relationship between the Fisher informationand information-theoreticquantities has been derived only recently (Clarke amp Barron 1990 Rissanen1996) in the limit of a large number of observations This link was exhib-ited rst in the context of parameter estimation (Clarke amp Barron 1990)for the case of statistically independent and identically distributed obser-vations Then it was generalized to a broader context within the frameworkof stochastic complexity with as a result a rened ldquominimum descriptionlengthrdquo criterion for model selection (Rissanen 1996)

The rst goal of this article is to show that within the framework ofinformation theory this link manifests itself very naturally in the context ofneural coding

In the limit of a large number of neurons coding for a low-dimensionalstimulus (population coding) the mutual information between the ac-tivities of the neuronal population and the stimulus becomes equal tothe mutual information between the stimulus and an efcient gaus-sian estimator under appropriate conditions detailed in section 3Here ldquoefcientrdquo means that the variance of this estimator reaches theCramer-Rao bound Since this variance is related to the Fisher infor-mation the equality provides a quantitative link between mutual andFisher informations

This equality is also shown to hold for a single cell in the case of agaussian noise with vanishing variance

The mutual informationbetween the stimulus and an efcientgaussianestimator reaches the mutual information between stimulus and theneuronal activities asymptotically from below

In the light of this relationship between Fisher and mutual informationwe examine in section 5 several issues related to population codes usingneurons coding for an angular variable with a triangular or bell-shapedtuning curve Such neurons are common in many neural structures Cellsof the postsubiculum (Taube Muller amp Ranck 1990) and anterior thalamicnuclei (Taube 1995) of the rat are tuned to its head direction Cells in MTcortex (Maunsell amp Van Essen 1983) of the monkey are tuned to the direction

Mutual Information Fisher Information and Population Coding 1733

of perceived motion Cells in motor cortex of the monkey (GeorgopoulosKalaska Caminiti amp Massey 1982) are tuned to the direction of the armWe study the case of an array of N neurons ring as a Poisson process inresponse to an angular stimulus with a frequency dened by the tuningcurve of the neuron in an interval of duration t In many cases Poisson pro-cesses are considered to be reasonable approximations of the ring processof cortical neurons (see eg Softky amp Koch 1993)

We calculate the Fisher information with an arbitrary density of pre-ferred angles Next we address the question of the optimization of the tun-ing curves making use of the link between mutual information and Fisherinformation The optimal density of preferred angles (ie the one that maxi-mizes the mutual information) is calculated as a function of the distributionof angles in section 52 As shown by Seung and Sompolinsky (1993) theFisher information in the large N limit diverges when the tuning width ofthe neurons goes to zero We show in section 53 that a nite tuning widthstems from optimization criteria which consider a nite system in whichonly a small number of spikes has been emitted by the whole populationWe illustrate our results using triangular tuning curves in section 54

2 General Framework

21 Parameter Estimation and Population Coding In the general con-text of parameter estimation one wishes to estimate a parameter h from aset of N observations fxi i D 1 Ng acute Ex (where the xirsquos might be discreteor continuous) h may characterize a model P(Ex|h ) which is expected tobe a good description of the stochastic process generating the observationsfxig In the simplest case the xirsquos are independent realizations of the samerandom variable and

P(Ex|h ) DNY

iD1

p(xi |h ) (21)

It may be the casemdashbut this is not necessarymdashthat the true process pcurren(x)belongs to the family under consideration so that pcurren(x) D p(x|ht) where htis the true value of the parameter

In the context of sensory coding and more specically population coding(see eg Seung amp Sompolinsky 1993 Snippe 1996) h is a stimulus (eg anangle) and the information about this stimulus is contained in the activitiesfxi i D 1 Ng of a population of a large number N of neurons In thesimplest case xi represents the activity of the ith neuron of the output layerof a feedforward network with no lateral connection so that the probabilitydensity function (pdf) P(Ex|h ) is factorized

P(Ex|h ) DNY

iD1

pi(xi |h ) (22)


Here pi(xi |h ) is the (neuron-dependent) pdf of the activity xi at neuron iwhen the input stimulus takes the value h

If the task of the neural system is to obtain a good estimate of the stimulusvalue the problem is a particular case of parameter estimation where thereexists a true valuemdashthe one that generated the observed activity Ex

22 The Cramer-Rao Bound In general one can nd different algo-rithms for computing an estimate Oh (Ex) of h from the observation of Ex Ifthe chosen estimator Oh (algorithm) is unbiased that is if

ZdNxP(Ex|h ) Oh (Ex) D h

the variance of the estimator

s2h D

D( Oh iexcl h )2

E

h

in which h ih denotes the integration over Ex given h (a sum in the case of adiscrete state vector) with the pdf P(Ex|h ) is bounded below according to(Cramer-Rao bound see eg Blahut 1988)

s2h cedil

1J (h )

(23)

where J (h ) is the Fisher information

J (h ) Dfrac12

iexcl 2 ln P(Ex|h ) h 2

frac34

h

(24)

For a multidimensional parameter equation (23) is replaced by an inequal-ity for the covariancematrix with J (h ) the Fisher informationmatrix beingthen expressed in terms of the second derivatives of ln P(Ex|h ) (Blahut 1988)For simplicity we will restrict the discussion to the case of a scalar param-eter and consider the straightforward extension to the multidimensionalcase in section 32

An efcient estimator is one that saturates the bound The maximumlikelihood (ML) estimator is known to be efcient in the large N limit

3 Mutual Information and Fisher Information

31 Main Result We now give the interpretation of the Cramer-Raobound in terms of information content First note that the Fisher informa-tion (see equation (24)) is not itself an information quantity The terminol-ogy comes from an intuitive interpretation of the bound our knowledge(ldquoinformationrdquo) about a stimulus h is limited according to this bound Thisqualitative statement has been turned into a quantitative statement inClarke


and Barron (1990) and Rissanen (1996) Here we give a different presentationbased on a standard information-theoretic point of view which is relevantfor sensory coding rather than from the point of view of parameter estima-tion and model selection

We consider the mutual information between the observable Ex and thestimulus h It can be dened very naturally in the context of sensory cod-ing because h is itself a random quantity generated with some pdf r (h )which characterizes the environment The mutual information is dened by(Blahut 1988)

I[h Ex] DZ

dhdNxr (h ) P(Ex|h ) logP(Ex|h )

Q(Ex) (31)

where Q(Ex) is the pdf of Ex

Q(Ex) DZ

dhr (h )P(Ex|h ) (32)

Other measures of the statistical dependency between input and outputcould be considered but the mutual information is the only one (up to amultiplicativeconstant) satisfying a set of fundamental requirements (Shan-non amp Weaver 1949)

Suppose there exists an unbiased efcient estimator Oh D T(Ex) It has meanh and variance 1J (h ) The amount of information gained about h in thecomputation of that estimator is

I[h Oh] D H[ Oh] iexclZ

dhr (h )H[ Oh |h ] (33)

where H[ Oh] is the entropy of the estimator

H[ Oh ] D iexclZ

d Oh Pr( Oh ) ln Pr( Oh )

and H[ Oh |h] its entropy given h The latter for each h is smaller than theentropy of a gaussian distribution with the same variance 1J (h ) Thisimplies

I[h Oh] cedil H[ Oh] iexclZ

dhr (h )12

lnsup3

2p eJ (h )

acute (34)

Since processing cannot increase information (see eg Blahut 1988 pp158ndash159) the information I[h Ex] conveyed by Ex abouth is at least equal to theone conveyed by the estimator I[h Ex] cedil I[h Oh] For the efcient estimatorthis means

I[h Ex] cedil H[ Oh] iexclZ

dhr (h )12

lnsup3

2p eJ (h )

acute (35)


In the limit in which the distribution of the estimator is sharply peakedaround its mean value (in particular this implies J (h ) Agrave 1) the entropy ofthe estimator becomes identical to the entropy of the stimulus The right-hand side (rhs) in the above inequality then becomes equal to IFisher plusterms of order 1J (h ) with IFisher dened as

IFisher D H(H) iexclZ

dhr (h )12

lnsup3

2p eJ (h )

acute (36)

In the above expression the rst term is the entropy of the stimulus

H(h ) D iexclZ

dhr (h ) ln r (h ) (37)

For a discrete distribution this would be the information gain resultingfrom a perfect knowledge of h The second term is the equivocation due tothe gaussian uctuations of the estimator around its mean value We thushave in this limit of a good estimator

I[h Ex] cedil IFisher (38)

The inequality (see equation 38) with IFisher given by equation 36 givesthe essence of the link between mutual information and Fisher informationIt results from an elementary application of the simple but fundamentaltheorem on information processing and of the Cramer-Rao bound

If the Cramer-Rao bound was to be understood as a statement on infor-mation content I[h Ex] could not be strictly larger than IFisher If not therewould be a way to extract from Ex more information than IFisher Hence theabove inequality would be in fact an equality that is

I[h Ex] D iexclZ

dhr (h ) ln r (h ) iexclZ

dhr (h )12

lnsup3

2p eJ (h )

acute (39)

However the fact that the equality should hold is not obvious The Cramer-Rao bound does not tell us whether knowledge on cumulants other thanthe variance could be obtained Indeed if the estimator has a nongaussiandistribution the inequality will be strictwe will give an example in section 4where we discuss the case of a single output cell (N D 1) In the large N limithowever there exists an efcient estimator (the maximum likelihood) andrelevant probability distributions become close to gaussian distributionsso that one can expect equation 39 to be true in that limit This is indeedthe case and what is proved in Rissanen (1996) within the framework ofstochastic complexity under suitable but not very restrictive hypotheses


In the appendix we show using completely different techniques thatequation 39 holds provided the following conditions are satised

1 All derivatives of G(Ex|h ) acute ln P(Ex|h )N with respect to the stimulus hare of order one

2 The cumulants (with respect to the distribution P(Ex|h )) of order n ofaG0

h C bG00h are of order 1Nniexcl1 for all a b n

The meaning of the second condition is that at a given value of N thecumulants should decrease sufciently rapidly with n This is in particulartrue when xi given h are independent as for model 22 but holds also in themore general case when the xi are correlated provided the above conditionshold as we show explicitly in the appendix using an example of correlatedxi

32 Extensions and Remarks

Multiparameter Case and Model Selection It is straightforward to extendequation 38 to the case of a K-dimensional stimulus Eh with pdf r ( Eh )and to derive the equality equation 39 for K iquest N The Fisher informationmatrix is dened as (Blahut 1988)

Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh

The quantity IFisher for the multidimensional case is then

IFisher D iexclZ

dKh r (Eh ) ln r (Eh ) iexclZ

dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)

The second term is now equal to the entropy of a gaussian with covariancematrix J iexcl1(Eh ) averaged over Eh with pdf r ( Eh ) In the large N limit (K lt ltN) one gets as for K D 1 the equality I D IFisher

One can note that formulas 39 and 310 are also meaningful in the moregeneral context of parameter estimation even when h is not a priori a ran-dom variable Within the Bayesian framework (Clarke amp Barron 1990) it isnatural to introduce a prior distribution on the parameter space r (h ) Typ-ically this distribution is chosen as the attest possible one that takes intoaccount any prior knowledge or constraint on the parameter space ThenI tells us how well h can be localized within the parameter space from theobservation of the data Ex

Within the framework of MDL (minimum description length) (Rissanen1996) the natural prior is the one that maximizes the mutual informationmdashthat is the one realizing the Shannon capacity Maximizing I D IFisher with


respect to r one nds that this optimal input distribution is given by thesquare root of the Fisher information

r (h ) Dp

J (h )R

dh 0p

J (h 0 )

(for the multidimensional case J in the above expression has to be replacedby det J ) This corresponds to the stimulus distribution for which the neuralsystem is best adapted

Biased Estimators The preceding discussion can be easily extended tothe case ofbiased estimators that is for estimators Oh with lt Oh gth D m(h ) 6D h The Cramer-Rao bound in such a case reads

s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)

This is a form of the bias-variance compromise One can thus write aninequality similar to equation 34 replacing J by J (dmdh )2 In the limitwhere the estimator is sharply peaked around its mean value m(h ) one hasr (h )dh raquo P( Oh )d Oh and Oh raquo m(h ) so that

H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|

Upon inserting H[ Oh] in the rhs of the inequality 34 the terms dmdh cancel

The bound equation 38 is thus also valid even when the known efcientestimator is biased

The Cramer-Rao bound can also be understood as a bound for the dis-criminability d0 used in psychophysics for characterizing performance in adiscrimination task between h and h C dh (see eg Green amp Swets 1966)As discussed in Seung and Sompolinsky (1993)

d0 middot dhp

J (h ) (312)

with equality for an efcient estimator and with d0 properly normalizedwith respect to the bias

d02 D

plusmndh dm

dh

sup22

s2h

(313)

4 The Case of a Single Neuron

41 A Continuous Neuron with Vanishing Output Noise We considerthe case of a single neuron characterized by a scalar output V with a deter-


ministic function of the input (stimulus) h plus some noise with a possiblystimulus-dependent variance

V D f (h ) C z sp

g(h ) (41)

where f and g are deterministic functions and s is a parameter giving thescale of the variance of the noise and z is a random variable with an arbitrary(that is not necessarily gaussian) distribution Q(z) with zero mean and unitvariance We are interested in the low noise limit s 0 It is not difcultto write the Fisher information J (h ) and the mutual information I[h V] inthe limit of vanishing s One gets for sufciently regular Q()

I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)

where H(Z) is the entropy of the z-distribution Q

H(Z) D iexclZ

dzQ(z) log Q(z) (43)

For the Fisher information one nds

J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that

IFisher[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)

If the noise distribution Q is the normal distribution one has H(Z) D12 log 2p e and the integral in equation 44 is equal to 1 so that one hasI D IFisher Otherwise one can easily check that I gt IFisher in agreement withthe general result (see equation 38)

42 Optimization of the Transfer Function The maximization of themutual information with respect to the choice of the transfer function fhas been studied in the case of a stimulus-independent additive noise thatis g acute 1 by Laughlin (1981) and Nadal and Parga (1994) The expressionfor the mutual information equation 42 with g D 1 has been computedby Nadal and Parga (1994) What is new here is the link with the Fisherinformation

The mutual information is maximized when f is chosen according to theldquoequalization rulerdquo that is when the (absolute value of) the derivative of fis equal to the pdf r the activity V is then uniformly distributed between


its min and max values In the more general case in which g depends on thestimulus the maximum of I is reached when Of dened by

Of 0 acute f 0 p

g

satises the equalization rule

Of D AZ h

dxr (x) C B (46)

where A and B are arbitrary given parameters (for g D 1 they dene themin and max values of f ) An interesting case is g D f which is relevantfor the analysis of a Poisson neuron in the large time limit (see the nextsubsection) In this case f 0 pg D 2

pf

0 and the maximum of I is reached

when the square root of f satises the equalization ruleThe fact that the mutual information is related to the Fisher information

in the case of a single neuron with vanishing noise means that maximizinginformation transfer is identical to minimizing the variance of reconstruc-tion error In fact two different qualitative lines of reasoning were known tolead to the equalization rule one related to information transfer (the outputV should have a uniform distribution see eg Laughlin 1981) and onerelated to reconstruction error (The slope of the transfer function shouldbe as large as possible in order to minimize this error and this with theconstraint that f is bounded leads to the compromise | f 0 | D r A large er-ror can be tolerated for rare events) We have shown here the formal linkbetween these two approaches using the link between mutual and Fisherinformation

43 A Poisson Neuron A related case is the one of a single neuron emit-ting spikes according to a Poisson process (in the next section we will con-sider a population of such neurons) The probability for observing k spikesin the interval [0 t] while the stimulus h is perceived is

p(k|h ) D(ordm(h )t)k

kexp(iexclordm(h )t) (47)

where the frequency ordm is assumed to be a deterministic function ordm(h ) (thetuning curve) of the stimulus h

h ordm D ordm(h ) (48)

If the stimulus is drawn randomly from a distribution r (h ) the frequencydistribution P (ordm) is given by

P (ordm) DZ

dhr (h ) d ( ordm iexcl ordm(h )) (49)


The information-processing ability of such model neuron has been studiedin great detail by Stein (1967) The results of interest here are as follows

At short times the mutual information between the stimulus and the cellactivity is at rst order in t (Stein 1967)

I(t) raquo tZ

dordmP (ordm)ordm logordm

macute I1(t) (410)

where m is the mean frequency One can easily check that I1(t) cedil I(t) for anyduration t In fact at long times information increases only as log t in thelarge time limit one gets (Stein 1967)

I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)

From this expression one gets that the optimal tuning curve is such thatpordm is uniformly distributed between its extreme values ordmmin and ordmmax We

can now analyze this result in view of the relationship between Fisher andmutual information Making the change of variable ordm h with

r (h )dh D P (ordm)dordm

together with equation 48 one can rewrite the mutual information at largetimes precisely as

I(t) D IFisher (412)

where IFisher is dened as in equation 36 with J (h ) the Fisher informationassociated with this single neuron

J (h ) D tordm

02(h )

ordm(h ) (413)

This result can be understood in the following way In the limit of large tthe distribution of the number of emitted spikes divided by t V acute k t tendsto be a gaussian with meanordm(h ) and varianceordm(h ) t so that the propertiesofthe spiking neuron become similar to those of a neuron having a continuousactivity V given by

h V D ordm(h ) C zp

ordm(h ) t

where z is a gaussian random variable with zero mean and unit varianceThis is a particular case of equation 41 with s D 1

pt f () D g() D ordm()


5 Population of Direction-Selective Spiking Neurons

51 Fisher Information We now illustrate the main statement of section3 in the context of population coding We consider a large number N ofneurons coding for a scalar stimulus (eg an angle) Equation 39 tells usthat to compute the mutual information we rst have to calculate the Fisherinformation

When the activities fxig of the neurons given h are independent P(Ex|h ) DPpi(xi |h ) the Fisher information can be written

J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)

where hiih is the integration over xi with the pdf pi(xi |h )We restrict ourselves to the case of neurons ring as a Poisson process

with rateordmi(h ) in response to a stimulush 2 [iexclp p ]ordmi(h ) therefore representthe tuning curve of neuron i We make the following assumptionsordmi(h ) hasa single maximum at the preferred stimulushi the tuning curve depends ononly the distance between the current stimulus and the preferred one andis a periodic function of this distance

ordmi(h ) D w (h iexclhi) (52)

through the same function w The locations of the preferred stimuli of theneurons are independently and identically distributed (iid) variables inthe interval h 2 [iexclp p ] with density r(h )

Since our model neurons re as a Poisson process the information con-tained in their spike trains in an interval of duration t is fully containedin the number of spikes xi emitted by each neuron in this interval For aPoisson process we have the law

pi (xi |h ) D(ordmi (h )t)xi

xiexp(iexclordmi(h )t) (53)

From equations 51 and 53 we can easily calculate the Fisher information

J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )

For N large we can replace the sum by the average over the distribution ofpreferred stimuli that is

J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )


Figure 1 Triangular tuning curve corresponding to a minimal frequency ordmmin D05 Hz ordmmax D 40 Hz a receptive eld half-width a D 40 degrees a preferredangle hi D 60 degrees

For an isotropic distribution r(h ) D 1(2p ) we recover the result of Seungand Sompolinsky (1993)

To understand how the Fisher information depends on other parametersof the tuning curve w we redene

w (h iexclhi) D fordmmin C (ordmmax iexcl ordmmin)Wsup3

|h iexclhi |

a

acute

where ordmmin and ordmmax are the minimal and maximal frequency a is the widthof the tuning curve and W is a decreasing function of |h iexcl hi | a such thatW D 1 for the preferred stimulus h D hi and W D 0 for stimuli far from thepreferred stimulus |h iexclhi | Agrave a In terms of these parameters we have

J (h ) D tN(ordmmax iexcl ordmmin)

a

Zdzr(h C az)

W0 (z)2

ordmminordmmaxiexclordmmin

C W(z)

The particular case of a triangular tuning curve

W(x) Draquo

(1 iexcl |x|) x 2 [iexcl1 1]0 |x| gt 1

(54)

is shown in Figure 1 It will be considered in more detail below For thistuning curve and for a uniform distribution of preferred stimuli the Fisherinformation has the simple form


p aln

ordmmax

ordmmin (55)


Thus as already noted by Seung and Sompolinsky (1993) the Fisherinformation diverges in different extreme cases when the maximal fre-quency ordmmax goes to innity and when the tuning width a goes to zeroMoreover functions W can be found such that the Fisher information di-verges (eg W (x) D

p1 iexcl x2) for any value of ordmmin ordmmax and a Thus the

optimizationof the Fisher information with respect to these parameters is anill-dened problem without additional constraints Note that in these casesthe equation relating the Fisher information to the mutual information is nolonger valid

There is however a well-dened optimization problem which is theoptimization with respect to the distribution of preferred orientations It isconsidered in section 52 Then we show how nite size effects transformthe problem of the optimization of both Fisher and mutual informationwith respect to the tuning width a into a well-dened problem Last wepresent some numerical estimates of these quantities inserting some realdata (Taube et al 1990) in equation 39

52 Optimization over the Distribution of Preferred Orientations Weask which distribution of preferred orientations r(h ) optimizes the mutualinformation I Obviously the optimal r will depend on the distribution oforientations r (h ) Optimizing equation 39 with respect to r(h 0 ) subject tothe normalization constraint

Rr(h 0 )dh 0 D 1 gives

Zdh

r (h )Rdh 00r(h 00 )y (h iexclh 00 )

y (h iexcl h 0 ) D ct for all h 0

in which we have dened

y (x) Dw 0 (x)2

w (x) (56)

This condition is satised when

r (h ) DR

dh 0r(h 0 )y (h iexclh 0)Rdh 0y (h 0 )

(57)

Thus the optimal distribution of preferred stimuli is the one that convolvedwith y (ie a quantity proportional to the Fisher information) matches thedistribution of stimuli Of course in the particular case of r (h ) D 1(2p ) weobtain ropt(h ) D 1(2p ) Note that equation 57 is also valid for unboundedstimulus values

This result (equation 57) is specic to the optimization of the mutualinformation Different results would be obtained for say the maximizationof the average of the Fisher information or the minimization of the averageof its inverse In fact there is no optimum for the mean Fisher informationsince it is linear in r()


Figure 2 (Left) SD of the reconstruction error after a single spike as a functionofa (Right) Mutual information between the spike and the stimulus as a functionof a Note that minimizing the SD of the reconstruction error is in this casedifferent from maximizing the mutual information

53 Finite Size Effects The Case of a Single Spike We have seen thatthe Fisher information in the large N limit diverges when the tuning widtha goes to zero To investigate whether this property is specic to the large Nlimit we study the case of a nite number of neurons in a very short timeinterval in which a single spike has been emitted by the whole populationin response to the stimulus h In this situation it is clear that the optimalestimator of the stimulus (the ML estimate in that case) is given by thepreferred stimulus of the neuron that emitted the spike For nite N theCramer-Rao bound in general is not saturated and we have to calculatedirectly the performance of the estimator It is a simple exercise to calculatethe standard deviation (SD) of the error made by such an estimate for atriangular tuning curve given in equation 54

SD(error) D

s4p 3ordmmin C a3(ordmmax iexcl ordmmin)6(2pordmmin C a(ordmmax iexclordmmin))

which always has a minimum for 0 lt a lt p We show in Figure 2 theSD of the reconstruction error after a single spike as a function of a forordmmaxordmmin D 80

It has a minimum for a about 50 degrees for which the SD of the error isabout 35 degrees


The mutual information on the other hand is

I D1

p Nordm

microa

ordmmax iexcl ordmmin

sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C

C (p iexcl a)ordmmin logplusmnordmmin

Nordm

sup2i

where

Nordm D ordmmin Ca

2p(ordmmax iexclordmmin)

It also has a maximum for positive a The width that maximizes I is differentfrom the width that minimizes the SD of the reconstruction error as shownin Figure 2 This is the case in general for nongaussian tuning curves Inthis case the half-width maximizing the mutual information is around 20degrees Note that in a wide range of a the rst spike brings about 2 bits ofinformation about the stimulus

Thus a nite optimal a stems from the constraint of already minimizingthe error when only a small number of spikes have been emitted by thewhole neuronal array It implies that the largest receptive elds are mostuseful at very short times when only a rough estimate is possible whilesmaller receptive elds will be most useful at larger times when a moreaccurate estimate can be obtained

54 Application to the Analysis of Empirical Data In this section weuse the experimental data of Taube et al (1990) to show how equation 39can be used to estimate both Fisher and mutual information conveyed bylarge populations of neurons on an angular stimulus (in this case the headdirection of a rat) Taube et al (1990) have shown that in the postsubiculumof rats tuning curves can be well tted by triangular tuning curves andthat the distribution of preferred orientations is consistent with a uniformdistribution They also determined the distribution of the parameters of thetuning curveordmmax a and the signal-to-noise ratio (SNR) a D ordmmaxordmmin overthe recorded neurons These data indicate that these parameters have animportant variability from neuron to neuron Equation 55 in the case ofsuch inhomogeneities has to be replaced by

J (h ) DtNp

Zdordmmaxdada Pr(ordmmax a a)

ordmmax

a

sup31 iexcl 1

a

acuteln a (58)

in which Pr(ordmmax a a) is the joint probability of parameters ordmmax a and a


Figure 3 (Left) Minimal reconstruction error as given by the Cramer-Rao boundfor N D 1000 (full curve) N D 5000 (dashed curve) postsubiculum neurons us-ing data from Taubeet al (1990)asa function of time (Right) Mutual informationfor N D 1000 (full curve) N D 5000 (dashed curve) using the same data andequation 310

Under global constraints one may expect each neuron to contribute in thesame way to the information that is (ordmmaxa)(1 iexcl 1a) ln a is constant Thiswould imply that the width a increases with ordmmax Figure 9 of Taube et al(1990) shows that there is indeed a trend for higher ring rate cells to havewider directional ring ranges

We can now insert the distributions of parameters measured in Taubeet al (1990) in equation 58 to estimate the minimal reconstruction errorthat can be done on the head direction using the output of N postsubicu-lum neurons during an interval of duration t It is shown in the left partof Figure 3 Since we assume that the number of neurons is large themutual information conveyed by this population can be estimated usingequation 39 It is shown in the right part of the same gure In the caseof N D 5000 neurons the error is as small as one degree even at t D 10ms an interval during which only a small proportion of selective neu-rons has emitted a spike Note that one degree is the order of magni-tude of the error made typically in perceptual discrimination tasks (seeeg Pouget amp Thorpe 1991) During the same interval the activity of thepopulation of neurons carries about 65 bits about the stimulus Doublingthe number of neurons or the duration of the interval divides the minimalreconstruction error by

p2 and increases the mutual information by 05

bit


6 Conclusion

In this article we have exhibited the link between Fisher information andmutual information in the context of neural coding This link was rst de-rived in the context of Bayesian parameter estimation by Clarke and Barron(1990) and then in the context of stochastic complexity by Rissanen (1996)We have shown that the result of Rissanen applies to population codingmdashthat is when the number of neurons is very large compared to the dimen-sion of the stimulus Our derivation of the link uses completely differenttechniques The result is that the mutual information between the neuralactivities and the stimulus is equal to the one between the stimulus and anideal gaussian unbiased estimator whose variance is equal to the inverseof the Fisher information The result is true not only for independent ob-servations but also for correlated activities (see Rissanen 1996 and theappendix) This is important in the context of neural coding since noise indifferent cells might in some cases be correlated due to common inputs orto lateral connections

This result implies that in the limit of a large number of neurons max-imization of the mutual information leads to optimal performance in theestimation of the stimulus We have thus considered the problem of opti-mizing the tuning curves by maximizing the mutual information over theparameters dening the tuning curves optimization of the choice of pre-ferred orientations widths of the tuning curves In the simple model wehave considered the optimal value for the width is zero as in Seung andSompolinsky (1993) However we have shown that nite size effects neces-sarily lead to a nonzero optimal value independent of the decoding scheme

We have discussed in detail the case of a one-dimensional stimulus (anangle) A similar relationship between mutual information and the Fisherinformation matrix holds for any dimensionality of the stimulus as long asit remains small compared to the number of neurons It would be straight-forward to consider in the more general case the optimization of the tuningcurves Zhang Ginzburg McNaughton and Sejnowski (1998) have com-puted the Fisher information matrix for two- and three- dimensional stim-uli Their results imply that optimal tuning curve parameters will dependstrongly on the dimensionality of the stimulus

We have briey discussed the cases of a nite number of neurons andthe short time limit In this case maximization of the mutual informationleads in general to different results than does minimization of the varianceof reconstruction error as found also in networks with the same number ofinput and output continuous neurons (Ruderman 1994) We are currentlyworking on these limits for which many aspects remain to be claried

We have not addressed the problem of decoding In the asymptotic limitthe maximum likelihood (ML) decoding is optimal Recently Pouget andZhang (1997) showed that a simple recurrent network can perform the com-putation of the ML estimate This suggests that the optimal performance


from the point of view of both information content and decoding can bereached by a simple cortical architecture

Appendix

Our goal is to derive equation 310 that is to compute the mutual informa-tion I D I[P r] between the random variables Ex and h working in the largeN limit We recall that Ex can be seen as either a set of N observations relatedto the measurement of an unknown parameter h or the set of responses ofN neurons to a stimulus h The mutual information I is dened by

I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ

dhr (h )P(Ex|h ) (A2)

In equation A1 h ih denotes the integration over Ex given h with the pdfP(Ex|h ) We dene

G(Ex|h ) acute1N

ln P(Ex|h ) (A3)

We will make the following hypothesis

1 All derivatives of G with respect to the stimulus h are of order 1 in thelarge N limit

2 The cumulants of order n of xG0h C yG00

h are of order 1Nniexcl1 in the largeN limit

Both properties are veried for the factorized models (see equations 21 and22) but also in some cases in which xi given h are correlated variables aswe show at the end of the appendix

The large N limit allows us to use the saddle-point method (Bhattacharyaamp Rao 1976 Parisi 1988) for the computation of integrals over h in par-ticular for the computation of the pdf Q(Ex) using the fact that P(Ex|h ) willappear to be sharply peaked around its most probable value the maximumlikelihood (ML) estimator of h We will use standard cumulant expansionsfor the integration over Ex in the equivocation part of I and this will eventu-ally lead to the announced result equation 310

Distribution of Ex The pdf Q(Ex) can be written

Q(Ex) DZ

dhr (h ) exp NG(Ex|h ) (A4)


For large N the integral is dominated by the maxima of the integrand Theseare dened by the solutions of

G0h (Ex|h ) D 0 (A5)

which satisfy G00h (Ex|h ) lt 0 Above we have denoted by G0

h (resp G00h) the rst

(resp second) partial derivative of G with respect to h Let us assume thatG(Ex|h ) has a single global maximum at hm(Ex) The Taylor expansion aroundhm(Ex) is

G(Ex|h ) D G(Ex|hm(Ex)) C12

G00h (Ex|hm(Ex))(h iexclhm(Ex))2 C

Using standard saddle-point techniques we nd

Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p

N|C(Ex)| exp [NGm(Ex)] (A7)

where

rm(Ex) acute r (hm(Ex)) (A8)

Gm(Ex) acute G(Ex|hm(Ex)) (A9)

and

C (Ex) acute G00h (Ex|hm(Ex)) (A10)

Note that hm(Ex) is the ML estimator of h

The Mutual Information Integration over h Let us start with the fol-lowing expression of the mutual information

I D iexclZ

dhr (h ) ln r (h ) CZ

dNx Q(Ex)Z

dh Q(h | Ex) ln Q(h |Ex)

with Q(h |Ex) D P(Ex|h )r (h)Q(Ex) The rst term is the entropy of the input distribu-

tion The second term can be written

iexclZ

dNx Q(Ex) ln Q(Ex) CZ

dNxZ

dh P(Ex|h )r (h ) ln P(Ex|h )r (h ) (A11)


In the above expression the rst part is the entropy of Ex in which we canreplace Q(Ex) by Qm(Ex) as given in equation A7 leading to

iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para

The last term in equation A11 can be written asZ

dNxZ

dh A(Ex|h ) exp A(Ex|h )

with

A(Ex|h ) acute ln P(Ex|h )r (h )

NowZ

dhA(Ex|h ) exp A(Ex|h ) D l

Zdh exp lA|lD1

which is again computed with the saddle-point method

ZdhA(Ex|h ) exp A(Ex|h ) D l

s2p

lN|C(Ex)|exp l [NGm C lnrm]

shyshyshyshyshylD1

D Qm

microNGm C lnrm iexcl

12

para

Finally putting everything together the mutual information can be writtenas

I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p

N |C (Ex)|exp [NGm(Ex)]

sup312

lnN |C (Ex)|

2p e

acute (A12)

It is interesting to compare equations 310 and A12 As in equation 310the rst term above is the entropy H[h ] D iexcl

Rdhr (h ) ln r (h ) of the stimulus

distribution the second term the equivocation is given in equation A12by the average over the pdf of Eu of the logarithm of the variance of theestimator

The Mutual Information Integration over Ex The last difculty is toperform in equation A12 the trace on Ex One cannot apply the saddle-pointmethod directly because the number of integration variables is preciselyequal to the number N that makes the exponential large However thedifculty is circumvented by the introduction of a small (compared to N)auxiliary integration variables in such a way that the integration over the


xirsquos can be done exactly Then we again use the fact that N is large to performthe integration over the auxilary variables to leading order in N

First we use the relation

F(hm(Ex)) DZ

dhF(h )|G00h (Ex|h )|d

iexclG0

h (Ex|h )cent

in order to deal with hm(Ex) which is valid for an arbitrary function F Wethen use an integral representation of the delta function

diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent

Similarly in order to deal with G00h (Ex|h ) we introduce conjugate variables t

Ot For any function F we can write

F(G00h (Ex|h )) D

Zdtd Ot

12p

F(t ) expiexcli Ot (t iexcl G00

h (Ex|h ))cent

Putting everything together we get

I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp

iexcli Ott C K(h y Ot )

cent (A13)

in which

K(h y Ot ) D lnfrac12

expsup3

iexcli Ot 2G(Ex|h )h 2

C iyG(Ex|h )

h

acute frac34

h

(recall that h ih DR

dNx exp[NG(Ex|h )] ) We now make the cumulantexpansion

laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2

h ) C cent cent centacute

for

A acute iexcli OtG00h C iyG0

h (A14)

The cumulant expansion will be valid if the cumulants of order n of Awith the law exp[NG(Ex|h )] decrease sufciently rapidly with n A sufcientcondition is

assumption the cumulants of order n of A (n D 1 2 )

are of order 1Nniexcl1 (A15)


Using the following identities obtained by deriving twice 1 D h1ih withrespect to h

0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets

K D i Ot J iexcl Ot 2D2

2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)

where J D Z are given by

J acute iexcllaquoG00

h

noth D N

DiexclG0

h

cent2E

h

D2 acute NplusmnDiexcl

G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth

Note that the Fisher information J (h ) is equal to N J and that D2 and Z areof order 1 because of the assumption A15

In these terms we have

I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3

i Ot (t C J) iexclOt 2D2

2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute

Our last task is to integrate over the remaining auxiliary variables t Ot y Using the fact that D2 iexcl Z2

J gt 0 deduced from the Schwartz inequality

lt G0h (G00

h iexcl lt G00h gt ) gt2 middot lt G02

h gt lt (G00h iexcl lt G00

h gt )2 gt

the integrations over y and Ot are simple gaussian integrations leading to

I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J

The integration over t is with a gaussian weight centered at t D iexclJ andwith a width going to zero as N goes to innity

limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)


Using the fact that the Fisher information is J (h ) D NJ we obtain

I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute(1 C O(1N)) (A17)

which is the announced result (equation 310)The conditions (see A15) of validity of the calculation are satised when

xi given h are independent as in equations 21 and 22 but can also besatised when they are correlated We discuss below these two cases

Conditional Independence of Activities In the case of independent neu-rons the model in equation 22 one can easily check that the cumulantexpansion at order n gives terms of order 1Nniexcl1 Indeed in that case onehas

G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X

iAi with Ai D iexcli Ot

2gi(xi |h )

h2 C iygi(xi |h )

h (A19)

The cumulant expansion then reads

laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i

Eiexcl hAii2) C O(1N3)

acute (A20)

Thus equation A16 holds with J D Z given by

J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)

Correlated Neurons The conditions on the cumulants of A A15 do notimply that the xi are independent but they do have the qualitative mean-ing that they convey of order N independent observations To see this we


give an example of correlated activities for which the conditions are satis-ed

We consider the following simple model Each xi can be expressed interms of the same N independent random variables ja a D 1 N as

xi DX

aMiaja (A22)

where M is a h-independent invertible matrix and the j rsquos are given h statistically independent variables of arbitrary pdf ra(j |h ) a D 1 NThe factorized case is recovered for M diagonal In the case where the rrsquosare gaussian and M is orthogonal equation A22 is the principal componentdecomposition of the xrsquos We show now that the case M invertible witharbitrary rrsquos satises the conditions A15

First it isobvious that the result (see equation 310) holdswith the changeof variables Ex Miexcl1 Ex D Ej one recovers the case of independent (givenh) activities One can then apply equation 310 to I(h Ej ) Since P(Ej |h ) DP(Ex|h )| det M| with M independent of h I(h Ex) D I(h Ej ) and the Fisherinformation associated to P(Ej |h ) is equal to the one associated to P(Ex|h ) sothat equation 310 holds for I(h Ex) Second one can check directly that theconditions A15 hold For our model G is

G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)

so that the cumulants of G0h (Ex|h ) and G00

h (Ex|h ) with respect to the pdf P(Ex|h ) areequal to the cumulants of G0

h (Ej |h ) and G00h (Ej |h ) with respect to the factorized

pdf P(Ej |h ) DQ

a ra(j |h ) for which A15 holds

Acknowledgments

We thank Alexandre Pouget and Sophie Deneve for an interesting discus-sion and Sid Wiener for drawing the data of Taube et al (1990) to ourattention We are grateful to Alexandre Pouget and Peter Latham for point-ing out a mistake in an earlier version of the article and to the referees forcomments that helped us to improve the article signicantly

References

Atick J J (1992) Could information theory provide an ecological theory ofsensory processing Network 3 213ndash251

Barlow H B Kaushal T P amp Mitchison G J (1989)Finding minimum entropycodes Neural Comp 1 412ndash423

Bhattacharya R N amp Rao R R (1976) Normal approximation and asymptoticexpansions New York Wiley


Bialek W Rieke F de Ruyter van Steveninck R amp Warland D (1991)Readinga neural code Science 252 1854ndash1857

Blahut RE (1988) Principles and practice of information theory Reading MAAddison-Wesley

Clarke B S amp Barron A R (1990) Information theoretic asymptotics of Bayesmethods IEEE Trans on Information Theory 36 453ndash471

Cover T M amp Thomas J A (1991) Information theory New York WileyGeorgopoulos A P Kalaska J F Caminiti R amp Massey J T (1982) On the

relations between the direction of two-dimensional arm movements and celldischarge in primate motor cortex J Neurosci 2 1527ndash1537

Green D M amp Swets J A (1966) Signal detection theory and psychophysics NewYork Wiley

Laughlin S B (1981)A simple coding procedure enhances a neuronrsquos informa-tion capacity Z Naturf C36 910ndash912

Linsker R (1988) Self-organization in a perceptual network Computer 21 105ndash117

Maunsell J H R amp Van Essen D C (1983) Functional properties of neuronsin middle temporal visual area of the macaque monkey I Selectivity forstimulus direction speed and orientation J Neurophysiol 49 1127ndash1147

Nadal J-P amp Parga N (1994) Nonlinear neurons in the low noise limit Afactorial code maximizes information transfer Network 5 565ndash581

Parisi G (1988) Statistical eld theory Reading MA Addison-WesleyPouget A amp Thorpe S J (1991)Connexionist models of orientation identica-

tion Connection Science 3 127ndash142Pouget A amp Zhang K (1997) Statistically efcient estimations using cortical

lateral connections In M C Moza M I Jordan amp T Petsche (Eds) Advancesin neural information processing systems 9 (pp 97ndash103) Cambridge MA MITpress

Rissanen J (1996) Fisher information and stochastic complexity IEEE Trans onInformation Theory 42 40ndash47

Ruderman D (1994) Designing receptive elds for highest delity Network 5147ndash155

Seung H S amp Sompolinsky H (1993) Simple models for reading neural pop-ulation codes PNAS USA 90 10749ndash10753

Shannon S E amp Weaver W (1949) The mathematical theory of communicationUrbana IL University of Illinois Press

Snippe H P (1996) Parameter extraction from population codes A criticalassesment Neural Comp 8 511ndash529

Softky W R amp Koch C (1993) The highly irregular ring of cortical cells isinconsistent with temporal integration of random EPSPs J Neurosci 13 334

Stein R (1967)The information capacity of nerve cells using a frequency codeBiophys J 7 797ndash826

Taube J S (1995) Head direction cells recorded in the anterior thalamic nucleiof freely moving rats J Neurosci 15 70ndash86

Taube J S Muller R U amp Ranck J B (1990) Head direction cells recordedfrom the postsubiculumin freely moving rats I Description and quantitativeanalysis J Neurosci 10 420ndash435


Van Hateren J H (1992) Theoretical predictions of spatiotemporal receptiveelds of y LMCS and experimental validation J Comp Physiology A 171157ndash170

Zhang K Ginzburg IMcNaughton BLamp Sejnowski T J (1998)Interpretingneuronal population activity by reconstruction A unied framework withapplication to hippocampal place cells J Neurophysiol 79 1017ndash1044

Received November 3 1997 accepted February 20 1998


A related domain which also belongs to information theory is the eldof statistical parameter estimation Here one typically has a sample of ob-servations drawn from a distribution that depends on a parameter or a setof parameters that one wishes to estimate The Cramer-Rao inequality thentells us that the mean squared error of any unbiased estimator of the under-lying parameter(s) is lower bounded by the inverse of a quantity which isdened as the Fisher information (Blahut 1988) This means that the Fisherinformation is a measure of how well one can estimate a parameter froman observation with a given probability law Thus in this sense it is also anldquoinformationrdquo quantity

In spite of the similar intuitive meanings of these two quantities an ex-plicit relationship between the Fisher informationand information-theoreticquantities has been derived only recently (Clarke amp Barron 1990 Rissanen1996) in the limit of a large number of observations This link was exhib-ited rst in the context of parameter estimation (Clarke amp Barron 1990)for the case of statistically independent and identically distributed obser-vations Then it was generalized to a broader context within the frameworkof stochastic complexity with as a result a rened ldquominimum descriptionlengthrdquo criterion for model selection (Rissanen 1996)

The rst goal of this article is to show that within the framework ofinformation theory this link manifests itself very naturally in the context ofneural coding

In the limit of a large number of neurons coding for a low-dimensionalstimulus (population coding) the mutual information between the ac-tivities of the neuronal population and the stimulus becomes equal tothe mutual information between the stimulus and an efcient gaus-sian estimator under appropriate conditions detailed in section 3Here ldquoefcientrdquo means that the variance of this estimator reaches theCramer-Rao bound Since this variance is related to the Fisher infor-mation the equality provides a quantitative link between mutual andFisher informations

This equality is also shown to hold for a single cell in the case of agaussian noise with vanishing variance

The mutual informationbetween the stimulus and an efcientgaussianestimator reaches the mutual information between stimulus and theneuronal activities asymptotically from below

In the light of this relationship between Fisher and mutual informationwe examine in section 5 several issues related to population codes usingneurons coding for an angular variable with a triangular or bell-shapedtuning curve Such neurons are common in many neural structures Cellsof the postsubiculum (Taube Muller amp Ranck 1990) and anterior thalamicnuclei (Taube 1995) of the rat are tuned to its head direction Cells in MTcortex (Maunsell amp Van Essen 1983) of the monkey are tuned to the direction




2 General Framework


P(Ex|h ) DNY

iD1

p(xi |h ) (21)



P(Ex|h ) DNY

iD1

pi(xi |h ) (22)







s2h D

D( Oh iexcl h )2

E

h


s2h cedil

1J (h )

(23)


J (h ) Dfrac12


frac34

h

(24)








I[h Ex] DZ


Q(Ex) (31)


Q(Ex) DZ

dhr (h )P(Ex|h ) (32)




dhr (h )H[ Oh |h ] (33)


H[ Oh ] D iexclZ




dhr (h )12

lnsup3

2p eJ (h )

acute (34)



dhr (h )12

lnsup3

2p eJ (h )

acute (35)




dhr (h )12

lnsup3

2p eJ (h )

acute (36)


H(h ) D iexclZ

dhr (h ) ln r (h ) (37)





I[h Ex] D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute (39)










Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































2 General Framework


P(Ex|h ) DNY

iD1

p(xi |h ) (21)



P(Ex|h ) DNY

iD1

pi(xi |h ) (22)







s2h D

D( Oh iexcl h )2

E

h


s2h cedil

1J (h )

(23)


J (h ) Dfrac12


frac34

h

(24)








I[h Ex] DZ


Q(Ex) (31)


Q(Ex) DZ

dhr (h )P(Ex|h ) (32)




dhr (h )H[ Oh |h ] (33)


H[ Oh ] D iexclZ




dhr (h )12

lnsup3

2p eJ (h )

acute (34)



dhr (h )12

lnsup3

2p eJ (h )

acute (35)




dhr (h )12

lnsup3

2p eJ (h )

acute (36)


H(h ) D iexclZ

dhr (h ) ln r (h ) (37)





I[h Ex] D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute (39)










Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References





































s2h D

D( Oh iexcl h )2

E

h


s2h cedil

1J (h )

(23)


J (h ) Dfrac12


frac34

h

(24)








I[h Ex] DZ


Q(Ex) (31)


Q(Ex) DZ

dhr (h )P(Ex|h ) (32)




dhr (h )H[ Oh |h ] (33)


H[ Oh ] D iexclZ




dhr (h )12

lnsup3

2p eJ (h )

acute (34)



dhr (h )12

lnsup3

2p eJ (h )

acute (35)




dhr (h )12

lnsup3

2p eJ (h )

acute (36)


H(h ) D iexclZ

dhr (h ) ln r (h ) (37)





I[h Ex] D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute (39)










Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































I[h Ex] DZ


Q(Ex) (31)


Q(Ex) DZ

dhr (h )P(Ex|h ) (32)




dhr (h )H[ Oh |h ] (33)


H[ Oh ] D iexclZ




dhr (h )12

lnsup3

2p eJ (h )

acute (34)



dhr (h )12

lnsup3

2p eJ (h )

acute (35)




dhr (h )12

lnsup3

2p eJ (h )

acute (36)


H(h ) D iexclZ

dhr (h ) ln r (h ) (37)





I[h Ex] D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute (39)










Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































dhr (h )12

lnsup3

2p eJ (h )

acute (36)


H(h ) D iexclZ

dhr (h ) ln r (h ) (37)





I[h Ex] D iexclZ


dhr (h )12

lnsup3

2p eJ (h )

acute (39)










Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References







































Jij

plusmnEh

sup2D

iexcl

2 ln Pplusmn

Ex| Ehsup2

hihj

+

Eh


IFisher D iexclZ


dKh r (Eh )12

lnsup3

(2p e)K

det J ( Eh )

acute (310)






r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































r (h ) Dp

J (h )R

dh 0p

J (h 0 )



s2hplusmn

dmdh

sup22 cedil1

J (h ) (311)


H[ Oh] D H[h] CZ

dhr (h ) log |dmdh

|




d0 middot dhp

J (h ) (312)


d02 D

plusmndh dm

dh

sup22

s2h

(313)





V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































V D f (h ) C z sp

g(h ) (41)


I[h V] D H(H) CZ

dhr (h )12

logf

02(h )s2g(h )

iexcl H(Z) (42)


H(Z) D iexclZ



J (h ) Df

02(h )s2g(h )

Zdz

Q02(z)

Q(z) (44)

so that


dhr (h )12

logf

02(h )s2g(h )

C12

logZ

dzQ

02(z)

Q(z)(45)






Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































Of 0 acute f 0 p

g


Of D AZ h

dxr (x) C B (46)


pf










P (ordm) DZ





I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































I(t) raquo tZ


macute I1(t) (410)


I(t) DZ

dordmP (ordm) log

sup3P (ordm)

r2p eordm

t

(411)







J (h ) D tordm

02(h )

ordm(h ) (413)


h V D ordm(h ) C zp

ordm(h ) t







J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References



































J (h ) DNX

iD1

1

p2i (xi |h )

sup3pi(xi |h )

h

acute2+

ih

(51)









J (h ) D tNX

iD1

ordm0i (h )2

ordmi(h )


J (h ) D tNZ p

iexclpdh 0r(h 0 )

w 0 (h iexclh 0 )2

w (h iexclh 0 )






|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References




































|h iexclhi |

a

acute



a

Zdzr(h C az)

W0 (z)2


C W(z)


W(x) Draquo


(54)



p aln

ordmmax

ordmmin (55)








Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References






































Zdh




y (x) Dw 0 (x)2

w (x) (56)


r (h ) DR


(57)






SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































SD(error) D






I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































I D1

p Nordm

microa


sup3ordm2

max

2log

plusmnordmmax

Nordm

sup2

iexclordm2

min

2log

plusmnordmmin

Nordm

sup2iexcl 1

4

plusmnordm2

max iexcl ordm2min

sup2C


Nordm

sup2i

where

Nordm D ordmmin Ca





J (h ) DtNp


ordmmax

a

sup31 iexcl 1

a

acuteln a (58)







bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References




































bit


6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References
































6 Conclusion








Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































Appendix


I DZ

dhr (h )frac12

lnP(Ex|h )Q(Ex)

frac34

h

(A1)


Q(Ex) DZ



G(Ex|h ) acute1N

ln P(Ex|h ) (A3)








Q(Ex) DZ




G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































G0h (Ex|h ) D 0 (A5)







Q(Ex) D Qm(Ex)sup3

1 C Osup3

1N

acuteacute (A6)

with

Qm(Ex) acute rm(Ex)

s2p


where



and




I D iexclZ


dNx Q(Ex)Z




iexclZ


dNxZ




iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































iexclZ

dNx Qm(Ex)micro

NGm C ln rm iexcl12

lnN|C (Ex)|

2p

para


dNxZ


with


NowZ


Zdh exp lA|lD1



s2p


shyshyshyshyshylD1

D Qm


12

para


I D iexclZ

dhr (h ) ln r (h )

CZ

dNx r (hm(Ex))

s2p


sup312

lnN |C (Ex)|

2p e

acute (A12)








F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































F(hm(Ex)) DZ


iexclG0

h (Ex|h )cent


diexclG0

h (Ex|h )cent

DZ

dy2p

expiexcliyG0

h (Ex|h )cent



F(G00h (Ex|h )) D

Zdtd Ot

12p


h (Ex|h ))cent


I D H[h] CZ

dhdydtd Otp

|t |p

N(2p )32

r (h )

poundsup3

12

lnsup3

N |t |2p e

acuteacuteexp


cent (A13)

in which


expsup3


C iyG(Ex|h )

h

acute frac34

h



laquoexp A

noth D exp

sup3hAih C

12

(DA2

E

hiexcl hAi2


for


h (A14)






0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































0 DlaquoG0

h

noth

0 DlaquoG00

h

noth

C NDiexcl

G0h

cent2E

h

one gets


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acute(A16)



h

noth D N

DiexclG0

h

cent2E

h


G00h

cent2E

hiexcl

laquoG00

h

not2h

sup2

Z acute NlaquoG0

h G00h

noth



I D H[h] CZ

dhdydtd Otp

|t |p

N(2p ) 32

r (h )sup3

12

lnsup3

N |t |

2p e

acuteacute

expsup3


2Niexcl y2J

2NC

y OtZN

C Osup3

1N2

acuteacute



lt G0h (G00



h gt )2 gt


I D H[h] CZ

dhr (h )

poundZ

dtp

2p

vuut N

D2 iexcl Z2

J

s|t |

J12

lnsup3

N |t |

2p e

acuteexp

sup3iexclN

2(t C J)2

D2 iexcl Z2

J


limN1

1p

2p

vuut N

D2 iexcl Z2

J

exp iexclN2

(t C J)2

D2 iexcl Z2

J

D d(t C J)



I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References

































I D iexclZ


dhr (h )12

lnsup3

2p eJ (h )





G(Ex|h ) D1N

X

igi(xi |h ) (A18)

so that

A D1N

X


2gi(xi |h )

h2 C iygi(xi |h )

h (A19)


laquoexp A

notD exp

X

ilog

frac12exp

Ai

N

frac34

D expX

i

sup31N

hAii C1

N2(DA2

i


acute (A20)


J D iexcl1N

X

i

frac122gi

h2

frac34

h

D1N

X

i

sup3gi

h

acute2+

h

D2 D1N

X

i

sup3sup32gi

h2

acute2+

h

iexclfrac12

2gi

h2

frac342

h

Z D1N

X

i

frac12gi

h

2gi

h 2

frac34

h

(A21)





xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References


































xi DX

aMiaja (A22)



G(Ex|h ) D iexcl1N

ln | det M| C1N

X

aln ra

sup3X

iMiexcl1

a i xishyshyh

(A23)




pdf P(Ej |h ) DQ


Acknowledgments


References































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