gelfand and smith (1990), read by

IntroductionSampling Approaches

ExamplesNumerical Illustrations

Conclusion

Sampling-Based Approaches to CalculatingMarginal Densities

ALAN E.GELFAND AND F.M.SMITH

Presented by Xiaolin CHENG

Reading Seminar, December 17, 2012

Xiaolin CHENG Sampling-Based Approaches to Calculating Marginal Densities



Conclusion

Contents

1 Introduction

2 Sampling ApproachesSubstitution AlgorithmSubstitution SamplingGibbs SamplingImportance-Sampling Algorithm

3 Examples

4 Numerical Illustrations

5 Conclusion




Conclusion

Introduction

AbstractThe problem addressed in this paper is how to obtain numerical esti-mates of available marginal densities, simply by means of simulatedsamples from available conditional distributions, and without recourseto sophisticated numerical analytic methods.




Conclusion

Introduction

We discuss and extend three alternative approaches put forward inthe literature for calculating marginal densities via sampling algo-rithms.

The Substitution Algorithm

The Gibbs Sampler Algorithm

The Importance-Sampling Algorithm




Conclusion

Substitution AlgorithmSubstitution SamplingGibbs SamplingImportance-Sampling Algorithm

Sampling Approaches

In relation to a collection of random variables,U1,U2, · · · ,Uk ,supposethat either

1 For i = 1, · · · , k , the conditional distributions Ui |Uj(j , i) areavailable,perhaps having for some i reduced forms Ui |Uj(j ∈ Si)(Si ⊂ {1, · · · , k })

2 The functional form of the joint density of U1,U2, · · · ,Uk isknown and at least one Ui |Uj(j , i) is available,

Where available means that samples of Ui can be straightforwardlyand efficiently generated.




Conclusion


Sampling Approaches

Densities are denoted generically by brackets and multiplication ofdensities is denoted by ∗, so

The joint distribution [X ,Y ]

The conditional distribution [X |Y ]

The marginal distribution [X ]

[X ,Y ] = [X |Y ] ∗ [Y ]∫h(Z ,W) ∗ [W ] to denote,for given Z, the expectation of the

function h(Z ,W) with respect to the marginal distribution forW.




Conclusion


Substitution Algorithm

The substitution algorithm for finding fixed-point solutions to certainclasses of integral equations is a standard mathematical tool thathas received considerable attention in the literature. Briefly review-ing the essence of their development using the notation introducedpreviously, we have

[X ] =

∫[X |Y ] ∗ [Y ] (1)




Conclusion



and[Y ] =

∫[Y |X ] ∗ [X ] (2)

so substituting (2) into (1) gives

[X ] =

∫[X |Y ] ∗

∫[Y |X ′] ∗ [X ′] =

∫h(X ,X ′) ∗ [X ′] (3)

where h(X ,X ′) =∫

[X |Y ] ∗ [Y |X ′] , with X ′ appearing as a dummyargument in (3),and of course [X ] = [X ′]




Conclusion



Now , suppose that on the right side of (3) , [X ′] were replaced by[X ]i , to be thought of as an estimate of [X ] = [X ′] arising at the ithstage of an iterative process. Then (3) implies that

[X ]i+1 =

∫h(X ,X ′) ∗ [X ′]i = Ih[X ]i

where Ih is the integral operator associated with h.




Conclusion



Exploiting standard theory of such integral operators , Tanner andWong ( 1987 ) showed that under mild regularity conditions this iter-ative process has the following properties( with obviously analogousresults for( [Y ] )

The true marginal density, [X ] , is the unique solution to (3)

For almost any [X ]0, the sequence [X ]1, [X ]2, . . . defined by[X ]i+1 = Ih[X ]i(i = 0, 1, . . .) converges monotonically in L1

to [X ]




Conclusion



Extending the substitution algorithm to three random variables X, Y,and Z , we may write [ analogous to (1) and (2) ]

[X ] =

∫[X ,Z |Y ] ∗ [Y ] (4)

[Y ] =

∫[Y ,X |Z ] ∗ [Z ] (5)

and[Z ] =

∫[Z ,Y |X ] ∗ [X ] (6)




Conclusion



Substitution of (6) into (5) and then (5) into (4) produces a fixed-point equation analogous to (3). A new h function arises with asso-ciated integral operator Ih , and these properties continue to hold inthis extended setting. Extension to k variables is straightforward.




Conclusion


Substitution Sampling

Returning to (1) and (2) , suppose that [X |Y ] and [Y |X ] are availablein the sense defined at the beginning.

For an arbitrary initial marginal distribution [X ]0 draw a singledistribution X0 from [X ]0

Given X0, since [Y |X ] is available draw Y (1) ∼ [Y |X (0)], andhence from (2) the marginal distribution of [Y (1)] is [Y ]1 =∫

[Y |X ] ∗ [X ]0

Now,complete a cycle by drawing X (1) ∼ [X |Y (1)]. Using (1),we then have X (1) ∼ [X ]1 =

∫[X |Y ] ∗ [Y ]1 =

∫h(X ,X ′) ∗

[X ′]0 = Ih[X ]0




Conclusion



Repetition of this cycle produces Y (2) and X (2), and eventually, af-ter i iterations, the pair (X (i),Y (i)) such that X (i) → X ∼ [X ], andY (i) → Y ∼ [Y ]. Repetition of this sequence m times each to theith iteration generates m iid pairs (X (i)

j ,Y (i)j ) (j = 1, . . . ,m). We call

this generation scheme substitution sampling.




Conclusion



If we terminate all repetitions at the ith iteration, the proposed den-sity estimate of [X ] (with an analogous expression for [Y ] ) is theMonte Carlo integration

[X ]i =1m

m∑j=1

[X |Y (i)j ] (7)




Conclusion



Extension of the substitution-sampling algorithm to more than tworandom variables is straightforward. We illustrate using the three-variable case.Paralleling (7), the density estimator of [X] becomes

[X ]i =1m

m∑j=1

[X |Y (i)j ,Z (i)

j ] (8)

with analogous expressions for estimating [Y] and [Z].




Conclusion



For k variables, U1, . . . ,Uk , the density estimator for [Us](s = 1, . . . , k)is

[Us]i =1m

m∑j=1

[Us |Ut = U(i)tj ; t , s] (9)




Conclusion


Gibbs Sampling

The Gibbs sampler has mainly been applied in the context of com-plex stochastic models involving very large numbers of variables,such as image reconstruction, neural networks, and expert system.




Conclusion


Gibbs Sampling

Algorithm

Given an arbitrary starting set of values U(0)1 ,U(0)

2 , . . . ,U(0)k

U(1)1 ∼ [U1|U

(0)2 , . . . ,U(0)

k ]

U(1)2 ∼ [U2|U

(1)1 ,U(0)

3 . . . ,U(0)k ]

U(1)3 ∼ [U3|U

(1)1 ,U(1)

2 ,U(0)4 , . . . ,U(0)

k ]...

U(1)k ∼ [Uk |U

(1)1 , . . . ,U(1)

k−1]




Conclusion


Gibbs Sampling

After i such iterations we would arrive at U(i)1 ,U(i)

2 , . . . ,U(i)k and we

have the following results

(U(i)1 ,U(i)

2 , . . . ,U(i)k )→ [U1, . . . ,Uk ] and hence for each s, U(i)

s →

Us ∼ [Us] as i → ∞.

Using the sup norm, rather than the L1 norm, the joint densityof (U(i)

1 ,U(i)2 , . . . ,U(i)

k ) converges to the true joint density

For any measurable function T of U1, . . . ,Uk whose expecta-tion exists

limj→∞

1i

i∑l=1

T(U(l)1 , . . . ,U(l)

k )→ E(T(U1, . . . ,Uk ))




Conclusion


Importance-Sampling Algorithm

Rubin(1987) suggested a noniterative Monte Carlo method for gen-erating marginal distributions using importance-sampling ideas andWe first present the basic idea in the two-variable case




Conclusion



suppose that

We seek the marginal distribution of X, given only the func-tional form of the joint density [X ,Y ] and the availability of theconditional distribution [X |Y ]

The marginal distribution of Y is not known

Choose an importance-sampling distribution for Y that has pos-itive support wherever [Y ] does and that has density [Y ]s

Then [X |Y ] ∗ [Y ]s provides an importance-sampling distribution for(X ,Y).




Conclusion



We draw iid pairs (Xl ,Yl) (l = 1, . . . ,N) from this joint distribution,for example, by drawing Yl from [Y ]s and Xl from [X |Yl]. Rubin’sidea is to calculate rl = [Xl ,Yl]/[Xl |Yl]∗ [Yl]s (l = 1, . . . ,N) and thenestimate the marginal density for [X ] by

[X ] =N∑

l=1

[X |Yl]rl/

N∑i=1

rl (10)




Conclusion



[X ] → [X ] with probability 1 as N → ∞ for almost every X. In ad-dition, if [Y |X ] is available we immediately have an estimate for themarginal distribution of Y : [Y ] =

∑Nl=1[Y |Xl]rl/

∑Nl=1 rl




Conclusion



The extension of the Rubin importance-sampling idea to the caseof k variables is clear. For instance, when k = 3, suppose thatwe seek the marginal distribution of X, given the functional formof [X ,Y ,Z ] and the availability of the full conditional [X |Y ,Z ]. Inthis case, the pair (Y ,Z) plays the role of Y in the two-variable casediscussed before, and in general we need to specify an importance-sampling distribution [Y ,Z ]s .




Conclusion



We draw iid triples (Xl ,Yl ,Zl) (l = 1, . . . ,N) and calculate rl =[Xl ,Yl ,Zl]/([Xl |Yl ,Zl] ∗ [Yl ,Zl]s). The marginal density estimate for[X ]

[X ] =N∑

l=1

[X |Yl ,Zl]rl/

N∑l=1

rl (11)




Conclusion

Examples

A major area of potential application of the methodology we havebeen discussed is in the calculation of marginal posterior densitieswithin a bayesian inference framework. In recent years, there havebeen many advances in numerical and analytic approximation tech-niques for such calculations, but implementation of these approach-es typically requires sophisticated numerical analytic expertise. Bycontrast, the sampling approaches we have discussed are straight-forward to implement.




Conclusion

Examples

Consider a general Bayesian hierarchical model having k stages. Inan obvious notation, we write the joint distribution of the data andparameters as

[Y |θ1] ∗ [θ1|θ2] ∗ [θ2|θ3] ∗ · · · ∗ [θk−1|θk ] ∗ [θk ] (12)

where we assume all components of prior specification to be avail-able for sampling. Primary interest is usually in the marginal poste-rior [θ1|Y ].




Conclusion

Examples

As a concrete illustration, consider an exchangeable poisson mod-el. Suppose that we observe independent counts, si , over differinglengths of time, ti (with resultant rate ρi = si/ti) (i = 1, . . . , p).Assume [si |λi] = P0(λi ti) and that the λi are iid from G(α, β) withdensity λα−1

i e−λi/β/βαΓ(α)




Conclusion

Examples

The parameter α is assumed known (in practice, we might treat αas a tuning parameter, or perhaps, in an empirical Bayes spirit, es-timate it from the marginal distribution of the s′i s), and β is assumedto arise from an inverse gamma distribution IG(γ, δ) with densityδγe−δ/βΓ(γ)




Conclusion

Examples

Letting Y = (s1, . . . , sp), the conditional distributions[λj |Y ] are sought.The full conditional distribution of λj is given by

[λj |Y , β, λi,i,j] = G(α + sj , (tj + 1/β)−1) (13)

whereas the full conditional distribution for β is given by

[β|Y , λ1, . . . , λp] = IG(γ + pα,∑

λi + δ) (14)




Conclusion

Examples

Given (λ(0)1 , λ

(0)2 , . . . , λ

(0)p , β(0)) the Gibbs sampler draw λ

(1)j ∼ G(α+

sj , (tj + 1/β(0))−1) (j = 1, . . . , p) and β(1) ∼ IG(γ+ pα,∑λ(1)i + δ) to

complete one cycle, generating (λ(i)1l , λ

(i)2l , . . . , λ

(i)pl , β

(i)l )(l = 1, . . . ,m)

the marginal density estimate for λj is

[ ˆλj |Y ] =1m

∑G(α + sj , (tj +

1

β(i)l

)−1)(j = 1, . . . , p) (15)

whereas

[ ˆβ|Y ] =1m

m∑l=1

IG(γ + αp,∑

λijl + δ)) (16)




Conclusion

Examples

Rubin’s importance-sampling algorithm is applicable in the setting(12) as well, taking a particularly simple form in the case k =2, 3. For k = 3, suppose that we seek [θ1|Y ]. The joint density[θ1, θ2, θ3|Y ] = [Y , θ1, θ2, θ3]/[Y ], where the functional form of thenumerator is given in (12).




Conclusion

Examples

We obtain the density estimator

ˆ[θ1|Y ] =∑

[θ1|Y , θ2l]rl/∑

rl

Returning to the exchangeable Poisson model, the estimator of themarginal density of λj under rubin’s importance-sampling algotithmis

ˆ[λj |Y ] =N∑

l=1

G(α + sj , (tj +1βj

)−1)rl/

N∑l=1

rl (17)

where rl = [Y |βl] ∗ [βl]/[βl |Y ]s




Conclusion

Numerical Illustrations

We apply the exchangeable Poisson model to data on pump fail-ures, where si is the number of failures and ti is the length of timein thousands of hours.




Conclusion

Pump system si ti ρi(×102)

1 5 94.320 5.32 1 15.720 6.43 5 62.880 8.04 14 125.760 11.15 3 5.240 57.36 19 31.440 60.47 1 1.048 95.48 1 1.048 95.49 4 2.096 191.010 22 10.480 209.9




Conclusion


Recalling the model structure and the forms of conditional distri-bution given by (13) and (14), we illustrate the use of the Gibbssampler for this data set, with p = 10, δ = 1, γ = 0.1, and for thepurposes of illustration α = ρ2/(Vρ − ρ

−1ρ∑p

i=1 t−1i ) Where ρ = α/β

and Vρ = p−1∑(ρi − ρ)2




Conclusion


The cycle is defined as follows:

draw initial β0 from [β]. where β ∼ IG(γ, δ)

draw independent λ(1)j from [λj |Y , β(0), λj , j , i]. which is a

G(α + sj , (tj + 1β(0)

)−1) distribution, j = 1, . . . , p

draw β(1) from [β|Y , λ(1)1 , . . . , λ(1)p ]. which is an IG(γ + αp, δ +∑

λ(1)i ) distribution.

Reinitialize the cycle with β1 and iterate, replicating each cycle mtimes.




Conclusion

Conclusion

We have emphasized providing a comparative review and explica-tion of three possible sampling approaches to the calculation of in-tractable marginal densities. The substitution, Gibbs, and importance-sampling algorithms are all straightforward to implement in severalfrequently occurring practical situation, thus avoiding complicatednumerical or analytic approximation exercises.


gelfand and smith (1990), read by

Education

available marginal densities

sampling algorithms

alternative approaches

ui uj j i

marginal densities alan

marginal distribution

marginal distribution

forms ui uj j si si