prior viability assessment for bayesian analysis

Journal of Statistical Planning and Inference 138 (2008) 1271–1286www.elsevier.com/locate/jspi

Prior viability assessment for Bayesian analysisMichael Goldstein, Allan Seheult∗

Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK

Received 20 January 2006; received in revised form 16 February 2007; accepted 24 April 2007Available online 22 May 2007

Abstract

We address the problem of determining whether the cost of a proposed Bayesian analysis is likely to be justified by the potentialbenefit. A method is described for identifying the likely order of magnitude benefits from the analysis, and this approach is appliedto an example concerning trading on the sugar market.© 2007 Elsevier B.V. All rights reserved.

Keywords: Bayes linear analysis; Commodity trading; Preposterior analysis; Price forecasting; Random walk; Temporal sure preference; Value ofanalysis

1. Introduction

We develop a method for assessing the practical viability of carrying out a Bayesian analysis. Such an analysis isconcerned with the combination of expert prior judgements with statistical models based on informative data in order toreduce certain key uncertainties in the application of interest. This type of analysis may be very expensive, and beforeembarking on such a program an organisation will need to estimate an order of magnitude budget for (i) data collectionand analysis; (ii) formulation and maintenance of the Bayesian model; (iii) expert elicitation for prior uncertainties;(iv) developing good solutions to the decision problems to which the uncertainties relate; (v) software requirements toallow the approach to be used routinely within the organisation. Such budget setting is a process which any organisationmust go through any time that they consider taking on a large Bayesian project, and is strongly project dependent. Ourobjective is to offer a method by which the organisation can get some idea as to whether the budget is likely to be wellspent.

In principle, to decide whether the budget is justified, we should weigh the costs of the analysis against the resultingbenefits. In many cases, the analysis will repay the investment only if it can sufficiently reduce posterior uncertaintyon certain key quantities to result in substantial improvements in some key operating procedures. If the project is largeand complex, then there will be considerable uncertainty as to the degree of success that is likely to be attained.

It is therefore useful, before such a potentially expensive Bayesian analysis is carried out, to make simple rule-of-thumb calculations to help us judge the likely benefits from the analysis. One approach is to construct a quickapproximate version of the modelling and elicitation, using easily available parts of the relevant data, to show conceptviability. However, the expense involved in constructing the simplified model may still be considerable, and there is no

∗ Corresponding author. Tel.: +44 191 3343113; fax: +44 191 3343051.E-mail addresses: [email protected] (M. Goldstein), [email protected] (A. Seheult).

0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jspi.2007.04.023

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1272 M. Goldstein, A. Seheult / Journal of Statistical Planning and Inference 138 (2008) 1271–1286

theoretical basis which would allow us to assess the extent to which final success may be predicted by the rough initialmodel, so that such simplifications may not address the decision support issues that we have raised.

In contrast to approaches which approximate the form of the Bayesian analysis, we suggest a viability assessmentbased on direct modelling of the objectives of the analysis. Our aim is to produce a simple order of magnitude assessmentof the relationship between the accuracy which is likely to be achieved by the analysis and the actual benefits whichwould derive from such an increase in accuracy. This assessment is based on a simple model for the potential gain ininformation whose effect on the operating procedures may be assessed both by simulation and by using sets of trainingdata. Our procedure is illustrated by an application based on our work on sugar trading.

The general methodology for constructing the prior viability analysis is described in Section 2. This methodologyworks by exploiting the temporal sure preference principle in order to construct the necessary stochastic relationshipsbetween the data outcomes and our inferences about these outcomes. In Section 3, we describe a sugar trading examplewhich we use to illustrate preliminary viability analysis. Specific application to a simple random walk model for sugarprices is analysed in detail and numerical evaluation of the viability analysis is provided. There is concluding discussionin Section 4.

2. Prior viability analysis

We now describe a general approach to Bayesian preliminary viability analysis. Our problem is as follows. Wemust decide whether to carry out a full Bayesian analysis intended to reduce uncertainty about a vector of randomquantities X.

The full analysis is likely to be costly in terms of data collection, model formulation, analysis, software creation andmaintenance, expert time in elicitation and so forth. We are uncertain as to how successful our analysis will be. One ofthe intended benefits of the reduction in uncertainty is improved performance for various tasks. Our aim is to considerwhether the potential benefits of the analysis outweigh the costs.

A full Bayesian analysis of the benefits of the analysis is likely to be extremely complex. We calculate a pragmaticlower bound on the likely benefit for the specified tasks from the Bayesian analysis, from which to gauge the viabilityof such an analysis.

We will not know how successful we will be in reducing uncertainty until we have carried out the full analysis, butwe may be able to make quantitative assessments, based on direct expert judgements or on simple modelling, as towhat levels of accuracy seem realistically attainable. By deriving a pragmatic lower bound on the value of any givenreduction in uncertainty, we identify the minimum reduction in uncertainty which offers a sufficient expected gain tooffset the cost of the analysis. If it seems plausible that we should be able to achieve such a reduction in uncertainty,then this offers a sensible rationale for carrying out the full Bayes procedure of modelling, elicitation, data collectionand so forth. There may be less tangible benefits from a full Bayesian analysis, and these should be separately assessed.

2.1. Prior judgements for future inferences

Our problem is as follows. If we carry out a Bayesian analysis, then the benefits that follow will arise because of theimprovement in our decision making. The decisions that we shall make will be functions of our posterior judgementsabout X given the Bayes analysis, while the gains from our decisions will be functions of the actual value of X. However,we have not yet decided which data we shall collect, how we shall model and analyse this data, or the level of detail towhich we shall elicit beliefs from experts. Therefore, we cannot construct a full Bayesian model for the analysis.

Therefore, we treat our posterior judgements as primitive quantities. We express directly our beliefs describing therelationship between X and our posterior judgements for X. Such assessments are similar to those that are made whenwe must assess the potential value of expert judgements; see Goldstein and O’Hagan (1996).

Thus, suppose that we envisage carrying out the full Bayesian analysis. If we carry out the analysis, we replace the

prior mean E[X] by a posterior mean E[X]. The value of E[X] is currently an unknown random vector. We will alsoreplace the prior variance matrix Var[X] with the random posterior variance matrix

Var[X] ≡ E[(X − E[X])(X − E[X])T].

M. Goldstein, A. Seheult / Journal of Statistical Planning and Inference 138 (2008) 1271–1286 1273

There is a simple relationship between current and future uncertainty judgements which derives from the temporal surepreference principle:

If you are sure that, at some future time, you will have a preference for W over U, where W and U are small,random money penalties, then you should not, at the present time, prefer U to W.

This principle was described, motivated and discussed in Goldstein (1997), where it is argued that, whatever methodyou choose to form your future beliefs, temporal sure preference implies that your current beliefs about your futurebeliefs must satisfy the stochastic relationship

X = E[X] + R, (1)

where the difference R = X − E[X] satisfies the conditions

E[R] = 0 and Cov[E[X], R] = 0. (2)

These conditions imply that

E[E[X]] = E[X]. (3)

Relations (1) and (3) give

Var[R] = E[(X − E[X])(X − E[X])T] = E[E[(X − E[X])(X − E[X])T]]= E[Var[X]] = EPV[X], (4)

where EPV[X] is our prior expectation for the posterior variance matrix. Thus,

Cov[X, E[X]] = Var[E[X]] = Var[X] − EPV[X]. (5)

Therefore, by specifying EPV[X], in addition to E[X] and Var[X], we may construct the full mean, variance andcovariance specification over X, E[X]. In our viability analysis, we use these relations to analyse the impact of differentchoices of EPV[X] on the potential gain from carrying out the full Bayes analysis.

2.2. Evaluating decisions based on posterior judgements

We now explain how to produce a viability function which translates each value of EPV[X] into a lower bound forthe corresponding expected gain for the tasks of interest. We shall use this function to make our judgement as to thevalue of EPV[X] which would be required in order to justify the cost of the analysis.

To assess how much we expect to gain by the full Bayes analysis, we require for comparison a baseline decisionprocedure which describes how we would proceed if we did not carry out such a full Bayes analysis. We also need toassess our prior probability distribution P[X] for X. For each value of X, we determine the gain g[X] (ideally in unitsof utility) that we would make by following our baseline decision procedure. Therefore, by simulation or by exactcalculation, we may evaluate our expected gain, g = E[g[X]], by making an immediate decision, where g is evaluatedwith respect to the distribution P[X].

We compare g with the expected gain if we carry out the full Bayes analysis before making our decision. The fullBayes calculation of this gain would be enormously complicated. However, we may derive a lower bound to this gainif we suppose that, rather than obtaining the full posterior probability distribution over X, we were, instead, only ableto discover the value of E[X]. Any price that we would consider it reasonable to pay for E[X] would be less than theprice that we would pay for the full posterior distribution, and thus serves as a lower bound for the value of the Bayesanalysis.

The value to us of learning E[X] will depend largely on how much the analysis may reduce our uncertainty about X.This uncertainty reduction is reflected in the value that we may specify for EPV[X]. Our aim is to assess the expectedbenefit of learning the value of E[X] for different values of EPV[X].

To assess this benefit, we must specify the decision choice that we will follow when we learn the value of E[X]. Anydecision rule that we use will give a lower bound for the expected gain from the optimal Bayes rule given E[X], andthus a lower bound for the value of the full Bayes analysis.


Suppose that we have chosen a decision procedure which depends only on the observed value of E[X] and thespecified value for EPV[X]. We suppose for now that there is a single decision, after which the value of X will berevealed and then we shall receive the payoff.

Therefore, our decision d will depend on the value that we learn for E[X] and the value we assign for EPV[X]; callthis choice of decision, d = d[E[X], EPV[X]]. Our gain will depend on the true value of X. Call this value G[X, d]. Wedefine

G[EPV[X]] = E[G[X, d]] (6)

to be the viability function over values of EPV[X].Therefore, we must assess our expectation for the gain function G[X, d] over values of X and E[X]. Suppose that

we specify a particular choice for EPV[X]. We do not have a full joint distribution for the pair X and E[X]. However,we have the marginal distribution for X, and from relations (1) and (5) we have a full mean, variance and covariancespecification for X and E[X]. We may exploit this specification by constructing an appropriate Bayes linear analysis.

2.3. Bayes linear methods

In general, the Bayes linear approach may either be motivated from a belief that the target quantities are roughlynormally distributed, or from the viewpoint that the Bayes linear approach is the appropriate form when dealing withpartial belief specifications based on means and variances, or as a pragmatic and tractable lower bound for expectedmean square error for an estimator using the full Bayes analysis, based on linear fitting. The Bayes linear analysisfollows directly from the formulation relating current and future beliefs through temporal sure preference; for anoverview of the Bayes linear approach, see Goldstein, 1999. In particular, if B and D are random vectors, then theadjusted expectation and variance for B given D are

ED[B] = E[B] + Cov[B, D]Var[D]−1(D − E[D]), (7)

VarD[B] = Var[B] − Cov[B, D]Var[D]−1Cov[D, B]. (8)

It follows that the adjusted expectation and variance for E[X] given X can be written

EX [E[X]] = (I − R[X])X + R[X] E[X], (9)

VarX [E[X]] = (I − R[X])R[X]Var[X], (10)

where

R[X] = EPV[X]Var[X]−1 (11)

is a dimensionless matrix with all its eigenvalues in (0, 1).These specifications provide the adjusted mean and variance for E[X], for each X. However, to assess the expected

gain function G[EPV[X]], we need a full probability distribution for E[X] given X. We therefore choose a plausibledistribution, with the given adjusted mean and adjusted variance for E[X], and carry out the analysis with this distribution.We may then explore the effects of using a range of distributions for E[X], to give more sensitivity as to the importantfurther aspects of our beliefs about posterior expectations in determining expected gains.

We now may compute, analytically or by simulation, the value of G[EPV[X]], for fixed EPV[X]. For example, wemay simulate values of X from P[X], and, for each such X, we may simulate values of E[X] from our chosen distributionwith mean and variance given by (9) and (10). For each pair of values of X and E[X], we evaluate d and therefore thevalue of G[X, d]. Averaging over simulations gives the value of G[EPV[X]].

The viability function G[EPV[X]] reveals how much we must expect to reduce variation over X in order to justify anygiven cost of analysis. The analysis is viable if sufficiently large gains correspond to reductions in uncertainty whichwe may be reasonably confident of achieving without too much cost. Alternatively, we may place prior probabilities onthe different values EPV[X], and thus of G[EPV[X]], that we may achieve, and therefore assess whether the expectedbenefit is justified by the expected cost. If the viability analysis suggests that it is unclear whether to continue, thenwe may choose to conduct a more careful version of the analysis, based on more detailed aspects of the uncertaintyspecification and more effective decision rules.


2.4. Extending the viability analysis

(i) Historical data: If we have a reasonably large number m of historical data sets X1, . . . , Xm judged to be ex-changeable with X, then there is a useful supporting calculation that we may make. For each i, we simulate the valueof G[Xi , d] as above. Then we may estimate G[EPV[X]] by

G[EPV[X]] = 1

m

m∑i=1

G[Xi , EPV[X]]. (12)

A comparison of the analysis of simulated and historical data is a useful diagnostic for those aspects of the viabilityassessment which rely on the assessment of the prior distribution for X. With sufficient historical data, we might evenuse (12) as a fast and cheap alternative to the full simulation based assessment of G[EPV[X]], as this method onlyrequires the specification of the prior mean and variance for X, which are used to evaluate (9) and (10), so that we donot need to specify a full prior distribution for X.

(ii) Several decisions: We have supposed above that there is a single decision involved in the viability analysis.Suppose instead that there are several decisions to be taken, in sequence, and that before each decision a subset of thevalues of the elements of X is revealed. Each element of X that we observe modifies our expectation for each remainingelement of X. We proceed as follows.

Initially, we specify E[X] and EPV[X] as the adjusted mean and variance for X. For each element of X that weobserve, we may further adjust the mean and variance for the remaining elements of X using (7) and (8), as we havemade a full covariance specification between all the elements of X and E[X]. Therefore, provided that at each stagewe restrict attention to decisions which are functions of the current adjusted mean and variance for the unobservedelements in X, then we may evaluate the expected gain exactly as for a single decision, and again this will provide alower bound for the expected gain over all possible choices of decision.

(iii) Several problems: So far, we have supposed that there is a single problem for which the Bayesian analysis will berelevant. It may be that there are a sequence of problems to which the analysis will apply. We may judge that variabilityin each problem differs simply by location and scale factors. In such cases, we may construct a parameter vector �say, of location and scale parameters, and specify a prior probability distribution over �. Then, for each simulation inderiving our viability function, we begin by drawing a value for �. We construct our viability analysis with the priorexpectation E[X] determined by the location parameters and prior variance Var[X] determined by the scale parameters.We consider EPV[X] to be the expected posterior variance matrix for X for some standard value of the prior variance,and scale the value of EPV[X] for each simulation by the corresponding multiplier for the scale parameters. The averageof the simulations will therefore provide our viability function as before.

2.5. Enlarging the set of uncertain quantities

In many problems, the number of random quantities whose values are relevant to the decisions that we must chooseis very large, as is the case in the example we discuss in Section 3. A viability analysis carried out in the way thatwe have suggested above may therefore become unwieldy, as it requires informed prior judgements as to the likelymagnitude of the elements of a large posterior variance matrix. Therefore, we may decide to simplify the viabilityanalysis, by supposing that, instead of receiving the full posterior expectation vector for all the quantities, we onlyreceive the posterior expectation for some aspects of the vector.

In particular, suppose that our decision depends on a large vector Y for which we have a prior mean and varianceE[Y] and Var[Y]. However, suppose that we are primarily concerned with learning about some smaller set X = LY oflinear combinations of the components of Y. We therefore consider our expected gain if we are informed of the valueof E[X], but given no other information about the value of E[Y]. Again, this analysis offers a lower bound for the fullposterior analysis, which will be informative for all elements of Y.

To identify the information about Y that we receive by learning the value of E[X], we write Y in the following form:

Y = EX[Y] + RY, (13)

where

E[RY] = 0, Var[RY] = VarX[Y] and Cov[X, RY] = 0.


The first term on the right in Eq. (13), namely EX[Y], is a linear function of X. Therefore, any reduction in variancefor X automatically translates into a corresponding reduction in variance for EX[Y], so that the values of E[EX[Y]],EPV[EX[Y]] are determined by the corresponding quantities for X.

The random quantity RY is uncorrelated with X. The least informative updating of beliefs about RY which followsfrom learning the value of E[X] would be to suppose that E[X] is uninformative for RY, and thus does not affect themean and variance for RY, so that E[RY] = 0, and EPV[RY] = VarX[Y]. Again, this is a lower bound for the expectedinformation that would be provided by the full Bayesian analysis. We complete the specification by setting the expectedposterior covariance to be equal to the prior covariance, namely EPC[X, RY] = 0.

We may now construct the combined vector U = (X, RY), so that U is a linear transform of Y. The values E[U],EPV[U] are determined as above. Thus, as EPV[U] is a function only of EPV[X] and our prior assessments, we maycarry out the viability analysis exactly as in Section 2.2, by evaluating the lower bound to expected gain for differentvalues of EPV[X].

2.6. Summary

We now summarise the main steps in carrying out a viability analysis.V1 Problem specification: We must specify

(i) the vector Y of random quantities whose values determine the gain for each decision;(ii) the vector X = LY of primary interest, in the sense that the viability analysis will give a lower bound for the value

of the Bayesian analysis for different choices of EPV[X];(iii) the default decision procedure d0 if no Bayesian analysis is to be carried out;(iv) the decision procedure d we are using to give a lower bound for the value of a full Bayesian analysis. The procedure

only depends on the posterior distribution of Y through its posterior expectation E[Y] and EPV[Y], the expectedvalue of the posterior variance.

V2 Belief specification and analysis:

(i) We specify the prior distribution for Y.(ii) We choose a specification for EPV[X].

(iii) From the prior variance matrix for Y, we construct the vector RY = Y − EX[Y]. Set E[RY] = E[RY] = 0.We convert Y to the linearly equivalent vector U = (X, RY), extend the specification to EPV[U] by EPV[RY] =Var[RY] and EPC[X, RY] = 0, and deduce the corresponding value of EPV[Y].We deduce the mean and variance specification for E[Y] and Y as

E[E[Y]] = E[Y] and Var[E[Y]] = Var[Y] − EPV[Y] = Cov[Y, E[Y]].(iv) We construct the adjusted expectation and variance for E[Y] given Y using Eqs. (9) and (10) with X replaced

by Y.(v) We evaluate the form of the decision procedure given E[Y].

V3 Viability analysis:

(i) Simulate a value of Y from its prior distribution.(ii) Evaluate the gain from the default decision procedure.

(iii) Evaluate the Bayes linear adjusted mean and variance for E[Y] given Y using V2 (v). Simulate a value for E[Y]from an appropriate distribution with this mean and variance. Choose the decision corresponding to these valuesof E[Y] and EPV[Y]. Evaluate the gain. In some circumstances, we may be able to write out explicitly the decisionfunction and associated gain, in which case we may evaluate the expected gain directly. [If there is a sequence ofdecisions, repeat this procedure sequentially, updating the mean and variance of E[Y] at each stage, and accumulategains.]

(iv) Repeat steps (i)–(iii) N times (N large). The difference between the average for step (ii) and the average for step(iii) is a lower bound for the expected gain of the Bayes analysis, for the given choice of EPV[X].


(v) By repeating the analysis for different choices of EPV[X], construct a picture of the minimal gains that we canexpect, depending on how much we expect to be able to reduce the uncertainty over X by carrying out a fullBayesian analysis. This allows an informed judgement as to whether these gains are likely to justify the cost ofthe analysis.

(vi) If we have extensive historical data on equivalent systems, then we may supplement the simulation analysis aboveby an exactly equivalent analysis based on the observed, rather than the simulated, Y values.

3. Example: sugar trading

We motivate and illustrate our approach with reference to a problem related to our work with C. Czarnikow SugarLtd, who are concerned with a wide range of financial activities within the sugar industry. Czarnikow were interestedin the relevance of Bayesian methodology to aspects of their services. The particular trading rule that we were askedto consider is confidential, and so we will present an example that is similar in spirit. All of the features of the actualcase study carried out for the client have been preserved, namely the data, the beliefs, the modelling, the evaluations,and the displays. The single change that we have made is to replace the actual trading rule with an alternative for whichthe nature of the analysis is essentially the same.

The problem that we shall consider is as follows. Sugar traders consult Czarnikow as to the best time to place theirsugar on the market. A trader will have a certain amount of sugar to sell. Czarnikow are experts in the analysis of thesugar trade, based, in part, on their knowledge of such features as likely production over future sugar harvests andthe political and economic factors that influence the rate at which sugar is likely to be released onto the internationalmarket. Optimal solution of this problem is a complex problem in Bayesian modelling, based on careful forecastingof future production and detailed assessment of the relationship between future sugar production and sugar price inorder to get good posterior forecasts of future sugar prices, followed by, for example, backward induction based onmaximising overall expected utility for the sugar trader to assess the optimal sale policy. While we cannot fully evaluatethe benefits of the Bayesian approach without carrying out such an analysis, it is prudent to try to obtain some order ofmagnitude assessments of the potential benefits of the Bayesian approach before embarking on such a complicated andpotentially expensive form of analysis. We shall therefore derive preliminary quantification of the potential benefits forsugar traders of using Bayesian methods to attempt to improve medium term forecasts for sugar prices. To help addressthis question, we have historical data on the closing price for sugar for each of 1560 days on the New York exchange.Table 1 shows the data, tabulated as 26 consecutive quarters (60 working days) of prices against which we can assessour procedures. We now describe how to carry out the viability summarised in Section 2.6, using the numbering there.

3.1. Problem specification

V1 (i) Czarnikow were concerned to develop decision procedures which exploited their expert knowledge about thelikely medium term future behaviour of sugar prices. For our example, we consider the price series Y for sugar overthe next 60 days. Specifically, we denote by Y1, . . . , Yn the daily closing prices of a pound of white sugar on the NewYork exchange over a period of “one quarter” of n = 60 working days.

V1 (ii) We will suppose that the Bayesian analysis will focus strictly on reducing uncertainty for the price X ≡ Y60on the final day. We will therefore need to quantify the potential increase in trading profits which would be achievedby various levels of reduction of uncertainty for Y60.

V1 (iii) We suppose that the trader intends to sell all of the sugar within the n = 60 days. If we do not carry out theBayes analysis, then our default decision procedure d0 is that the trader will sell out on the first day. Item (B) in Section3.5 discusses other non-Bayesian default rules.

V1 (iv) Our alternative trading rule d for the viability analysis is as follows. Initially, Czarnikow make a forecastof Y60, the sugar price which will be obtainable in three months time. If the prior expectation for Y60 is lower thanthe initial sugar price Y0, then all the sugar is sold immediately; otherwise, no sugar is sold. Each day, the currentsugar price Yt is monitored and beliefs about Y60 are updated by Bayes linear adjustment on Y1 . . . , Yt and our futureposterior expectation E[Y60]. If on day t the adjusted expectation for Y60 is smaller than the observed value of Yt , thenall the sugar is sold; otherwise, all of the sugar is sold on day 60 at price Y60.

We have chosen this rule for simplicity of exposition, as it depends explicitly on only a single prior elicitation, and isstraightforward to implement, so that we may derive a closed form expression for the viability analysis. Also, it would

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1271–

1286Table 1Daily sugar prices (in US cents per lb) on the number 11 exchange for 25 quarters Q1, Q2, . . . , Q25 (each of 60 working days); and estimates of the simple random walk mean � and incrementvariance �2

z for each quarter derived from the prices for the immediately preceding quarter. The quarter preceding the first quarter is not shown

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25�2

z 0.02 0.05 0.03 0.04 0.05 0.09 0.08 0.03 0.04 0.02 0.04 0.02 0.01 0.02 0.01 0.02 0.01 0.03 0.03 0.04 0.02 0.03 0.05 0.02 0.02� 12.17 12.08 12.66 14.60 14.56 11.58 10.97 10.92 12.53 11.26 11.82 10.37 10.64 11.04 11.67 12.30 11.23 8.43 8.41 7.62 8.42 5.10 5.52 6.84 6.12

1 11.84 12.14 12.73 15.01 14.30 11.88 10.81 10.97 11.65 11.01 11.85 10.35 10.78 11.11 11.58 12.39 11.20 8.50 8.44 7.75 7.69 5.07 5.40 6.68 6.102 11.95 11.57 12.57 15.13 13.87 11.93 10.75 11.30 11.70 10.95 11.85 10.46 10.92 11.05 11.38 12.43 11.12 8.47 8.53 7.66 7.52 5.13 5.31 6.72 5.773 11.96 11.61 12.58 14.85 13.88 11.79 10.76 11.38 11.72 11.02 11.65 10.69 11.25 11.01 11.53 12.29 10.89 8.49 8.52 7.74 7.47 5.10 5.42 6.72 5.814 11.85 11.54 12.53 14.94 13.89 11.46 10.86 11.29 11.73 11.21 11.74 10.68 11.33 10.97 11.51 12.32 10.71 8.53 8.84 7.75 7.10 5.14 5.52 6.89 5.775 11.05 11.49 12.68 14.79 14.11 11.57 10.79 11.32 11.74 11.47 11.73 10.59 11.18 10.98 11.49 12.25 10.78 8.96 8.76 7.63 6.97 5.05 5.47 6.88 5.846 11.28 11.86 12.54 14.96 13.95 11.60 10.77 11.26 11.70 11.42 11.73 10.65 10.92 11.05 11.55 12.25 10.88 9.05 8.73 7.55 6.73 4.78 5.54 6.81 5.837 11.21 11.67 12.51 15.02 14.10 11.66 10.81 11.44 12.16 11.35 11.78 10.69 10.80 11.15 11.86 12.18 10.97 9.14 8.71 7.76 6.65 4.82 5.77 6.81 5.678 11.20 11.67 12.69 15.11 14.18 11.65 10.81 11.38 12.09 11.60 12.18 10.75 10.90 11.06 11.82 12.35 11.01 9.24 8.85 7.70 6.92 4.80 5.62 6.69 5.459 10.56 11.26 12.61 14.91 14.18 11.85 10.75 11.46 12.16 11.58 12.12 10.66 11.05 11.11 11.84 12.27 11.00 9.35 8.83 7.71 7.11 4.72 5.84 6.56 5.4910 10.57 11.32 12.52 14.89 14.20 12.02 10.83 11.37 12.23 11.44 12.01 10.66 11.32 11.16 11.69 12.34 10.75 9.18 8.94 7.60 6.76 4.58 5.79 6.46 5.4511 10.88 11.47 12.42 15.17 14.48 12.10 10.96 11.40 12.17 11.80 12.00 10.27 10.84 11.18 11.60 12.34 10.58 9.24 9.06 7.87 6.92 4.90 5.73 6.69 5.2912 10.91 11.52 12.45 15.39 14.23 12.62 11.18 11.43 12.41 11.74 12.03 10.30 10.88 11.21 11.79 12.31 10.59 9.09 9.05 8.05 6.81 5.34 5.76 6.94 5.4913 10.97 11.54 12.41 15.61 14.32 12.81 11.22 11.29 12.46 11.68 12.05 10.31 10.86 11.17 11.66 12.26 10.37 8.95 8.66 8.02 6.69 4.50 6.04 6.92 5.5614 11.01 11.70 12.47 15.61 14.28 12.83 11.23 11.38 12.48 11.67 11.96 10.40 10.93 11.34 11.79 12.31 9.82 8.85 8.71 7.95 6.71 4.61 6.14 6.88 5.4315 10.87 12.05 12.42 15.74 14.39 12.35 11.04 11.50 12.24 11.67 11.89 10.30 10.97 11.48 11.72 12.00 9.73 8.96 8.54 7.88 6.84 4.82 5.98 7.25 5.2916 10.77 12.09 12.15 15.45 14.27 12.31 11.25 11.54 12.25 11.54 11.64 10.35 11.03 11.51 11.68 12.03 9.94 8.99 8.59 8.05 6.83 4.77 6.03 6.89 5.3217 11.21 12.02 12.37 15.42 13.88 12.03 11.52 11.58 12.07 11.59 11.57 10.42 10.92 11.41 11.60 12.15 9.62 8.89 8.58 8.38 6.89 4.70 6.04 6.91 5.4018 11.66 11.88 12.29 15.18 14.04 10.79 11.70 11.54 11.90 11.75 11.43 10.46 10.77 11.45 11.70 12.22 9.64 8.56 8.53 8.38 6.67 4.88 6.11 7.01 5.4119 11.68 11.90 12.57 15.21 14.01 10.77 11.29 11.56 11.86 12.03 11.40 10.73 10.76 11.45 11.60 12.24 9.98 8.34 8.49 8.37 6.75 4.82 6.20 6.88 5.4720 11.77 11.86 12.82 14.26 13.70 10.25 10.25 11.60 11.68 11.84 11.42 10.56 10.90 11.37 11.61 12.49 9.85 8.26 8.19 8.35 6.63 4.82 5.99 7.09 5.4221 11.65 11.78 12.72 14.02 13.37 10.34 10.39 11.83 11.79 11.91 11.19 10.77 10.92 11.26 11.54 12.45 9.81 8.28 7.93 8.49 6.65 4.76 6.06 6.94 5.3922 11.29 11.72 12.79 14.42 13.28 10.38 10.41 11.84 11.55 12.17 11.30 10.86 10.88 11.49 11.51 12.38 9.73 8.08 7.93 8.46 6.68 4.90 6.13 6.96 5.5023 11.53 11.72 12.78 14.67 12.65 9.72 10.55 11.72 11.54 12.61 11.17 10.71 10.75 11.42 11.50 12.38 9.73 8.26 7.53 8.33 6.75 4.63 6.08 6.97 5.6124 11.50 11.82 12.70 14.78 12.60 9.87 10.47 11.89 11.77 12.43 11.21 10.69 10.90 11.36 11.40 12.44 9.50 8.25 7.50 8.48 6.81 4.62 5.88 7.00 5.7925 11.65 11.68 12.71 15.13 12.75 9.69 10.53 11.94 11.89 11.32 11.11 10.75 10.83 11.34 11.56 12.35 9.39 8.13 7.56 8.24 6.79 4.72 6.01 6.88 5.6626 11.82 11.70 12.70 14.93 12.82 9.72 10.43 11.73 11.91 11.38 11.04 10.73 10.92 11.36 11.49 12.33 9.34 8.07 7.55 8.03 6.73 4.85 6.05 6.90 5.5327 11.72 11.48 12.86 14.80 13.09 9.78 10.69 11.79 11.58 11.39 11.47 10.89 10.86 11.37 11.32 12.25 9.25 7.86 7.38 8.13 6.87 4.89 6.00 6.98 5.4928 11.94 11.54 12.89 14.53 13.24 10.05 10.66 11.67 11.63 11.69 10.78 10.99 10.81 11.06 11.29 12.10 9.40 7.55 7.70 8.23 6.34 4.82 6.08 6.81 5.5229 11.83 11.93 12.83 14.46 12.99 10.03 10.62 10.87 11.74 11.67 10.89 11.00 10.81 11.13 11.31 12.14 9.44 7.36 7.54 8.31 6.02 4.78 6.10 6.78 5.4930 12.26 11.88 12.80 14.37 12.82 10.25 10.61 11.09 11.62 11.64 10.85 10.94 10.79 11.08 11.22 12.17 9.63 7.71 7.59 8.30 5.97 4.79 6.09 6.71 5.2131 12.17 11.80 12.90 14.37 12.89 10.73 10.52 11.30 11.67 11.68 10.93 11.08 11.00 11.15 11.17 11.93 9.78 7.45 7.35 8.29 5.85 4.77 6.01 6.56 5.1332 11.98 11.82 13.21 14.12 13.02 10.61 10.56 11.49 11.51 11.77 10.88 11.07 10.96 11.11 11.05 12.16 9.85 7.59 7.21 8.24 5.50 5.00 6.09 6.23 4.9533 11.86 11.91 13.18 14.25 13.35 10.54 10.68 11.56 11.54 11.85 10.82 10.96 11.00 10.92 10.98 12.31 9.88 7.60 7.00 8.10 5.72 5.24 6.10 6.04 5.0734 11.97 11.82 13.19 14.44 11.76 10.55 10.67 11.76 11.15 11.65 10.78 10.87 11.00 11.00 10.76 12.29 9.83 7.98 7.10 8.20 5.72 5.49 6.07 5.91 5.0335 12.23 11.90 13.27 14.35 11.58 10.65 10.68 11.67 11.09 11.79 10.88 10.72 11.00 11.24 10.61 12.24 9.93 7.99 7.03 8.24 5.81 5.30 6.16 5.91 5.1436 12.21 11.81 13.15 13.74 11.61 10.50 10.94 11.72 11.11 11.66 10.81 10.64 11.04 11.05 10.89 12.21 9.83 7.97 7.15 8.25 6.04 5.34 6.70 6.03 5.1537 11.91 11.89 13.15 13.86 11.81 10.58 10.81 12.19 11.06 11.65 10.86 10.49 11.31 11.18 11.04 12.29 9.80 7.89 7.31 8.19 5.91 5.45 6.82 6.07 5.1238 12.17 11.85 13.26 13.92 11.71 10.56 10.60 12.48 11.09 11.51 10.73 10.51 11.37 11.06 11.17 12.36 9.77 7.84 6.95 8.12 5.90 5.41 6.65 6.06 4.9439 11.93 12.11 13.64 14.07 11.78 10.75 10.53 12.36 10.88 11.45 10.70 10.62 11.33 11.06 11.69 12.30 9.86 7.89 6.67 8.12 5.86 5.26 6.78 5.78 4.70

M.G

oldstein,A.Seheult/JournalofStatisticalP

lanningand

Inference138

(2008)1271

–1286

127940 11.85 12.18 13.66 14.25 11.54 10.70 10.55 12.15 10.54 11.69 10.77 10.57 11.24 11.11 11.87 12.22 10.19 8.22 6.91 8.09 5.82 5.26 6.90 5.76 4.6541 12.01 12.11 13.64 14.28 11.56 10.94 10.56 11.93 10.50 11.73 10.64 10.52 11.13 11.06 11.90 12.02 9.95 8.63 7.19 7.78 5.79 5.41 7.09 5.75 5.0842 11.79 12.03 13.58 14.52 11.18 10.79 10.64 12.04 10.51 11.90 10.57 10.35 10.95 11.09 11.81 11.96 9.94 8.74 7.10 7.58 5.72 5.34 7.06 5.98 4.9643 12.04 12.00 13.74 14.40 11.47 11.01 10.68 12.19 10.33 11.88 10.63 10.17 10.96 11.12 11.79 11.82 9.91 8.46 6.98 7.52 5.54 5.29 6.97 6.23 5.0544 12.19 12.11 13.80 14.58 11.56 11.04 10.77 12.00 10.40 11.87 10.65 10.17 11.29 10.97 11.90 11.86 9.87 8.40 7.04 7.58 5.59 5.29 6.70 6.07 5.0445 12.02 11.96 13.98 14.80 11.88 11.14 10.85 12.02 10.51 11.70 10.66 10.15 11.24 11.30 11.97 11.79 9.78 8.58 6.93 7.53 5.60 5.52 6.79 6.03 5.0746 12.09 12.06 14.59 14.79 11.86 11.15 10.87 12.12 10.59 11.78 10.70 10.24 11.27 11.16 11.94 11.46 9.51 8.54 7.04 7.62 5.47 5.85 6.70 6.10 5.1247 12.00 12.12 14.49 14.65 11.85 11.00 10.75 12.01 10.62 11.71 10.54 10.44 11.53 11.28 11.85 11.51 9.53 8.40 7.13 7.58 5.56 5.82 6.81 6.03 5.1648 12.14 12.14 15.16 14.98 11.61 10.99 10.58 12.29 10.87 11.73 10.49 10.41 11.46 11.26 11.80 11.54 9.30 8.34 7.62 7.83 5.70 5.98 6.77 6.02 5.2249 12.46 12.05 14.70 15.05 11.60 11.08 10.64 12.20 10.78 11.64 10.47 10.33 11.55 11.59 11.85 11.17 9.43 8.39 7.35 7.88 5.74 5.92 6.91 5.88 5.4250 12.42 12.04 14.71 14.75 11.57 11.45 10.74 12.31 10.76 11.48 10.30 10.39 11.49 11.49 11.78 11.20 9.35 8.64 7.36 7.86 5.82 5.95 6.93 5.74 5.2851 12.44 11.90 14.44 14.70 11.77 11.46 10.74 12.49 10.80 11.37 10.36 10.45 11.32 11.56 11.68 11.12 9.11 8.76 7.27 8.18 5.91 6.09 6.79 5.75 5.2552 12.36 12.38 14.29 14.73 11.67 11.34 10.83 12.66 10.91 11.48 10.35 10.42 11.14 11.63 11.85 11.28 9.02 8.67 7.48 8.74 5.59 5.95 6.95 5.88 5.3453 12.37 12.28 14.40 14.66 11.40 11.58 10.93 12.62 10.93 11.59 10.33 10.44 10.70 11.70 11.74 11.27 9.05 8.66 7.27 8.79 5.62 5.72 6.98 5.96 5.2754 12.33 12.26 14.71 14.59 11.22 11.39 10.84 12.59 11.05 11.88 10.40 10.58 10.80 11.66 11.73 11.23 9.15 8.71 7.60 8.80 5.63 5.60 6.94 5.92 5.1355 12.40 12.30 14.61 14.55 11.72 11.30 10.89 12.63 11.42 11.81 10.48 10.66 10.79 11.58 11.73 11.18 9.19 8.95 7.66 8.71 5.50 6.29 6.61 5.89 5.2056 12.36 12.54 14.46 14.35 11.61 10.71 10.79 12.61 11.50 11.66 10.53 10.60 10.88 11.68 11.81 11.03 9.15 9.00 7.66 8.75 5.44 6.13 6.68 5.88 5.2457 12.06 12.75 14.33 14.15 11.43 10.84 10.86 12.80 11.39 11.64 10.55 10.51 10.76 11.65 11.89 11.23 9.15 8.51 7.78 8.67 5.43 5.74 6.99 5.92 5.4558 12.10 12.82 14.41 14.34 11.52 10.63 10.97 13.03 11.38 11.72 10.38 10.63 10.84 11.58 11.78 11.23 9.04 8.45 7.72 8.70 5.25 5.37 6.99 5.92 5.3859 11.83 12.65 14.71 14.39 11.78 10.86 10.81 13.16 11.22 11.81 10.48 10.66 10.88 11.61 12.00 11.23 8.86 8.63 7.90 8.55 5.14 5.64 6.91 6.14 5.4760 12.08 12.66 14.60 14.56 11.58 10.97 10.92 12.53 11.26 11.82 10.37 10.64 11.04 11.67 12.30 11.23 8.43 8.41 7.62 8.42 5.10 5.52 6.84 6.12 5.41


be simple to explain and motivate to a trader, as it is based on a very simple heuristic, namely to hold if the price isexpected to rise and sell if the price is expected to fall. Further, this rule is not dependent on deep modelling of thefuture form of the series and so, if viability can be demonstrated for this procedure, then we may be fairly confidentthat the benefits from whichever trading rule emerges from a full Bayesian analysis should prove at least as substantial.Item (C) of Section 3.5 provides a brief discussion of more elaborate rules.

This completes the problem specification V1.

3.2. Belief specification

V2 (i) Data analysis data analysis for many 60 day historical sequences suggested a simple random walk for dailysugar price. Whether the random walk model is appropriate for modelling sugar prices is a controversial question, butthe model does give a simple basis for simulation, sufficient to demonstrate the general approach. Further discussionand comment on the appropriateness of this simple model can be found in item (A) in Section 3.5. We could similarlyconstruct a viability analysis for any alternative stochastic model for Yt .

We model the Yt as a random walk

Yt = � +t∑

i=1

Zi, t = 1, . . . , n, (14)

where the Zi are uncorrelated, zero-mean random quantities, each with variance �2z . Our prior beliefs about � (equiva-

lently Y0), �2z (reflecting market volatility) and the random walk model for Y1, . . . , Yn may be assessed from historical

data.V2 (ii) Our prior beliefs for X ≡ Y60 follow from the random walk model in (14); in particular, E[Yn] = � and

Var[Yn] = n�2z .

Following the discussion in Section 2, E[Yn] is our (random) posterior expectation for Yn. To simplify subsequentexpressions, we will write Yn for E[Yn]. The stochastic relationship resulting from the temporal sure preference principlein Section 2 becomes

Yn = Yn + Rn, (15)

where E[Rn] = 0 and Cov[Yn, Rn] = 0. Hence,

E[Yn] = E[Yn] = � and Var[Yn] = Var[Yn] − Var[Rn] = n�2z − EPV[Yn].

Denoting the variance ratio R[Yn] = EPV[Yn]/Var[Yn] in (11) by R, we can write EPV[Yn] = R n�2z and hence,

Var[Yn] = (1 − R)n�2z .

Thus, for each combination of � and �2z , the viability function will be determined by R ∈ (0, 1).

V2 (iii) We now construct the residual vector RY = Y − EYn [Y], since the mean and variance assessments of the Yt

are required for the trading rule d.It follows from the random walk model in (14) that

EYn [Yt ] = � + Cov[Yt , Yn]Var[Yn]−1(Yn − �) = qt� + ptYn, (16)

where pt = t/n and qt = 1 − pt . Hence,

EYn [Y] = qn� + pnYn, (17)

where pn = (p1, . . . , pn)T and qn = (q1, . . . , qn)

T.Therefore, as Y = EYn [Y] + RY = qn� + pnYn + RY and EPV[RY] = Var[RY], it follows that

EPV[Y] = Var[RY] + pnpTnEPV[Yn] = Var[RY] + pnpT

nnR�2z (18)


and

Var[RY] =[KCCTKT 0

0T 0

]�2

z ,

where K = [In−1| − pn−1/n], C is the n × n cumulative sum matrix, In−1 is (n − 1) × (n − 1) identity matrix and 0is the (n − 1)-vector of zeros.

As E[RY] = 0, we see that E[E[Y]] = E[Y] and

Var[E[Y]] = Var[Y] − EPV[Y] = pnpTnnR�2

z = Cov[Y, E[Y]]. (19)

V2 (iv) It is straightforward to show from (18) and (19) that

EY[Yn] = EYn [Yn] = (1 − R)Yn + R�, (20)

VarY[Yn] = VarYn [Yn] = nR(1 − R)�2z . (21)

We can extend these results to the adjusted expectation and variance for E[Y] given Y using (9) and (10) with E[Y]=�1,R(Y) in (11) determined from (18), Var[Y] = CCT�2

z and Cov[Y, E[Y]] given in (19).V2 (v) The Bayesian trading rule d = d(Yn,Y[t]), to sell or hold the sugar stock on day t, will be based on the Bayes

linear forecast

E(Yn,Y[t])[Yn] (22)

of the final price Yn adjusted by Yn and the prices Y[t] = (Y1, . . . , Yt )T

observed up to and including day t. In ourexample, d is such that we sell on day t if

Y1 < E(Yn,Y[1])[Yn], Y2 < E(Yn,Y[2])[Yn], . . . , Yt−1 < E(Yn,Y[t−1])[Yn] and Yt �E(Yn,Y[t])[Yn]

otherwise, hold the current stock over until the next day. Note that if we have not sold by day n−1 we must sell at Yn

on day n.We now write

Zi = EYn [Zi] + RZi, (23)

where RZiand Yn are uncorrelated and RZi

and Yn are uncorrelated. It then follows that

Cov[Zi, Yn] = Cov[EYn [Zi], Yn] = Cov[Zi, Yn]Var[Yn]−1Var[Yn] = (1 − R)�2z . (24)

We now compute (22). The following adjusted expectations, variances and covariances of Z1, . . . , Zn can be straight-forwardly derived using (24)

EYn[Zi] = (Yn − �)/n,

CovYn[Zi, Zj ] = (�ij − �n)�

2z ,

where �n = (1 − R)/n and �ij = 1 if i = j and 0 otherwise.As the Yt are partial sums of the Zi , we readily deduce that

EYn[Yt ] = qt� + pt Yn,

CovYn[Ys, Yt ] = min(s, t)[1 − �n max(s, t)]�2

z .


It is then straightforward to show that

CovYn[Yn,Y[t]] = (1 − R)�2

zaTt ,

�2z[VarYn

[Y[t]]]−1 = A−1t − A−1

t aTt (atA

−1t at − (1 − R)−1)−1aT

t A−1t ,

where at = (1, 2, . . . , t)T and A−1t = �t�T

t − eteTt , where the t × (t + 1) matrix �t takes successive first-differences

of the elements of any vector of length t + 1, and et = (0, . . . , 0, 1)T. These are all of the results needed to calculatethe expressions for the adjusted expectation of Yn given in (25) and the adjusted variance of Yn given in (26) requiredfor the Bayes trading rule d given in (27) below.

E(Yn,Y[t])[Yn] and corresponding variance Var(Yn,Y[t])[Yn] are given by

E(Yn,Y[t])[Yn] = � + qt (Yn − �) + R(Yt − �)

qt + Rpt

, (25)

Var(Yn,Y[t])[Yn] =[

Rqt

qt + Rpt

]n�2

z . (26)

Re-arranging (25), we sell at Yt on day t if we have not sold previously and

Yn �R� + (1 − R)Yt . (27)

otherwise, hold. We note that d only depends on Y[t] through Yt . Item (D) in Section 3.5 discusses this result and therole of sufficiency as applied to the random walk model.

We have now completed the belief specification V2.

3.3. Viability analysis

Our simulation procedure is as follows.V3 (i) We fix a value of R. We simulate a series y = (y1, . . . , yn) of daily prices at times t = 1, . . . , n from the

random walk model with the values of � and �2z simulated from a prior distribution assessed either subjectively or using

historical data.V3 (ii) The gain from the default decision rule d0 is y1.V3 (iii) It is readily seen from (27) that the probability distribution for the actual selling price SP is given by

Pr[SP = yt ] =

⎧⎪⎨⎪⎩

Pr[W �y1], t = 1,

Pr[max{y1, . . . , yt−1} < W �yt ], 2� t �n − 1,

Pr[W > max{y1, . . . , yn−1}], t = n,

(28)

where W = (Yn − R�)/(1 − R).Notice that some of the probabilities in (28) can be zero. It is straightforward to show from (20) that the expectation

and variance of W adjusted by the observed value Yn = yn are EYn [W ] = yn and VarYn [W ] = nR�2z/(1 − R); and if we

assume a particular distribution for Yn, such as Gaussian, we may evaluate the probability distribution in (28).The gain using d compared with d0 is computed as

Gy(R) =n∑

t=1

yt Pr[SP = yt ] − y1, R ∈ [0, 1] (29)

is the viability of the simple Bayes selling rule for a particular value of R for this y using whatever values of � and�2

z were chosen for the simulation. When R is “small”, we expect the rule to produce better gains over the strategyof selling on the first day; for example, R = 0 corresponds to knowing Yn, while R = 1 corresponds to receiving noadditional information about Yn beyond that implied by the prior mean and variance structure of the assumed randomwalk model for daily prices.


0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Relative expected posterior variance R

Gain distribution estimate from 1000 simulated quarters

Expecte

d v

alu

e a

nd q

uantile

s o

f

gain

dis

trib

ution in c

ents

per

lb

Fig. 1. Viability function: expected value (solid) and selected quantiles—5% (dot-dash), 25% (dot), 50% (dash), 75% (dot), 95% (dot-dash)—of theselling price distribution of d less the first day price Y1 of a quarter (in cents per lb) as a function of the relative expected posterior variance R ofthe last day price Y60, estimated by pooling the corresponding distributions for 1000 simulated quarters. Note that the 5%, 25% and 50% quantilescoincide for R ∈ [0, 0.3] as do the 25% and 50% quantiles for R ∈ [0.3, 1.00].

V3 (iv) We now repeat items (1), (ii) and (iii) a large number of times to obtain the average G(R) of Gy(R) over thesimulated values of y, � and �2

z .V3 (v) We now repeat (i)–(iv) to obtain the viability function G(R) for a grid of values of R.V3 (vi) We may supplement the simulation analysis by repeating the above calculations using a collection of historical

time series.We have now described all of the calculations required to carry out the viability analysis V3.

3.4. Viability analysis for the given data

We now apply the foregoing theoretical development using historical data on sugar prices to assess the viability ofthe simple Bayes trading rule determined by (27). The historical data, which are shown in Table 1, are daily closingprices for sugar (in US cents per lb) on the New York exchange for m = 25 quarters, each of 60 working days. Weuse these data to illustrate both simulation and historical data viability analysis, as described in Sections 2.2 and 2.4,respectively.

Viability based on simulated prices: We first generated m=1000 simulated random walk price series of length n=60as follows:

(a) The mean � was drawn from a Gaussian distribution with expectation 10 and variance 9 and then the precision 1/�2z

was drawn independently from a Gamma distribution with shape 1.7 and rate 0.025. This joint prior distributionwas assessed using the data in Table 1. Item (E) in Section 3.5 gives more detail on this assessment.

(b) The series was then generated from (14) by simulating Z1, . . . , Z60 independently from a Gaussian distributionwith expectation zero and the variance �2

z generated in (a) and then adding the � generated in (a) to each of the60 values.

(c) Next, the SP distribution in (28) for d was computed for each of the m=1000 simulated quarters. Fig. 1 shows theexpectation (solid line) and selected quantiles of the overall SP distribution obtained by pooling the distributionsin (28) generated from the 1000 simulated quarters. Note the following points.


0.0 0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Relative expected posterior variance R

Estimate of viability distribution from 25 quarters

Expecte

d v

alu

e a

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Fig. 2. Viability function: expected value (solid) and selected quantiles—5% (dot-dash), 25% (dot), 50% (dash), 75% (dot), 95% (dot-dash)—of theselling price distribution less the first day price Y1 of a quarter (in cents per lb) as a function of the relative expected posterior variance R of the lastday price Y60, pooled over the corresponding distributions for the 25 quarters. Quantiles coincide similarly to those in Fig. 1.

(i) The expectation curve, the average of the 1000 gain curves is our prior viability assessment of d; and, as wewould expect, it decreases as R increases. This curve indicates how much we expect to gain over first dayprice Y1 as a function of our ability to assess our uncertainty about Y60; for example, reducing our uncertaintyby 40% (R = 0.6) leads to a substantial average gain of about one-third of a cent per lb of sugar.

(ii) The distribution is skewed towards large values for all values of R, but more so for small values of R, as wewould expect.

(iii) The 5%, 25% and 50% quantile curves coincide for some intervals of R: 5%, 25% and 50% in [0, 0.3] and25% and 50% in [0.3, 1.00].

(iv) The 50% quantile is zero for all R, as it should be (at least approximately for the simulation) because for arandom walk the probability that Y60 is less than the starting price Y1 (so that we sell immediately withoutgain) must be 0.5. Also, the probability of non-negative gain will decrease from 1 at R = 0 (no uncertaintyabout Y60) to 0.5 at R = 1, while the probability of negative gain increases from 0 at R = 0 to 0.25 at R = 1.

(c) We may view the distribution summaries in Fig. 1 as an approximation to those for an infinite number of suchquarters.

Viability based on historical prices: We now give a brief account of an analysis described in Section 2.4 (i). Standardanalyses suggest that a random walk model is a plausible description for the sugar price series shown in Table 1.Also shown in Table 1 are the estimates � = y60 and �2

z = ∑60t=2(yt − yt−1)

2/59 of the simple random walk mean� and volatility variance �2

z for each quarter, based on the prices y1, . . . , y60 for the immediately preceding quarter.Prices for the quarter preceding the first quarter (Q1), not shown in Table 1, are used solely to estimate � and volatilityvariance �2

z for Q1. These estimates are taken to be our prior assessments for � and �2z for each of the m = 25

quarters.Fig. 2 shows selected quantiles and the expectation (solid line) of the overall SP distribution obtained by averaging

the distribution in (28) for each quarter over the 25 quarters. Although less smooth, the plot tells essentially the “samestory” as the corresponding plot in Fig. 1 based on simulation, providing further support for the appropriateness of theviability analysis.


3.5. Further notes on the example

(A) We have chosen a random walk model with constant variance (as an approximation to slowly varying volatility)for price rather than for the (often preferred) logarithm of price, primarily because our data analysis suggested thesmall benefits for the later were offset by the ease of interpretation and prior elicitation for the former. However,the viability calculations carry through for log price as a random walk, either via simulation or using historicaldata.While we could allow for a more intricate variance model of the Zt to be assessed, we believe that such an extralayer of modelling will not be a major factor in the current viability analysis. A careful Bayes analysis might welllead us to develop a much more detailed probability model for prices. However, for the prior viability assessmentthere is little value, even were it possible, in trying to average over all the models that we might develop, and wesuspect, in any case, that the changes in the expected posterior variance structure will be small.

(B) We might use a more sophisticated default rule, such as waiting until the price is a certain percentage above somerecent historical average price before selling. However, any such rule would be based on the supposition that thereis a simple, automatic way of beating the market. We would expect any such rule to be apparent to all traders inthe market, so that many traders would take this position (as you do not need to possess sugar to sell sugar on thecommodity market), so that the market should already hold the alternative price. The question as to whether themarket prices commodities fairly is, of course, somewhat controversial, but the view is widely held, and seems areasonable point of comparison for our analysis.

(C) The method that we describe would be straightforward in principle to apply to more elaborate rules based onsplitting the amount sold on different days, according to some utility based criterion, based on more detailedaspects of our prior specification, subject only to the increased complexity of the simulation procedures.

(D) Observe that the adjusted expectation of Yn in (25) depends on Yt = (Y1, . . . , Yt ) only through Yt . This suf-ficiency property is a consequence of certain basic properties of Bayes linear separation. We say that randomvector B separates random vectors A and C, written [A@C]/B if the adjusted covariance of A and C given B iszero, or equivalently if Cov[A − EB[A], C − EB[C]] = 0. Goldstein, 1990 shows that belief separation may beviewed as a generalised conditional independence property; in particular, it is shown that [B@(C, D)]/E implies[B@C]/(D, E), for any four vectors B, C, D and E.As Y1, . . . , Yn is a random walk, [Yn@Yt−1]/Yt so that Cov[Yn,Yt−1 − EYt [Yt−1]] = 0. Hence, by our in-formation assumption in Section 2.5, Cov[Yn,Yt−1 − EYt [Yt−1]] = 0, so that [Yn@Yt−1]/Yt and therefore[(Yn, Yn)@Yt−1]/Yt . Hence, by the generalised conditional independence property [Yn@Yt−1]/(Yt , Yn), andthe result follows.

(E) A priori, we chose � and the precision 1/�2z to be independent with Gaussian and Gamma distributions, respectively.

The data in Table 1 were used to assess the values of the parameters in these two distributions. The mean andstandard deviation of the values of � for the 25 quarters, shown at the head of the columns in Table 1, wereused as a basis to assess the mean and the standard deviation of the prior distribution for �. The parameters in theprior distribution for the precision 1/�2

z were assessed by applying a parametric empirical Bayes calculation to theindependent successive squared differences (Yi−Yi−1)

2 ∼ �2z �2

1; that is, the parameters in the Gamma distributionwere estimated by maximising the likelihood generated by the product of the marginal density functions of thesuccessive squared differences. The quality of the fit to the empirical distribution of the squared successivedifferences to the fitted marginal distribution (not shown) is very good.

4. Conclusions

We have described a general approach to assessing the potential viability for a Bayesian analysis, and shown how thisapproach may be applied in an example on commodity trading. While the approach must always be carefully matchedto the problem at hand, we feel that the ideas involved in such a viability analysis are widely applicable. We thereforesuggest that, for any problem where a proper Bayesian analysis would incur substantial expense, an attempt is made todevelop order of magnitude assessments of the potential benefits of the analysis, along the lines that we have suggested,both to avoid carrying out analyses where an unrealistic level of success would appear to be required in order to justifythe cost, and also to identify and support those analyses where good returns are likely to follow from more modestlevels of success. While our example was carried out in terms of monetary gain, there is no difficulty in making a similar


viability assessment based on general expected utility gains. In a similar way, each aspect of the viability analysis maybe more carefully modelled and evaluated, subject only to the constraint that the chosen procedures should not incurso large a cost as to cast doubt on the financial viability of the preliminary viability analysis itself.

Acknowledgements

We would like to thank Peter Thompson, Chris Pack and John Kovacs of C. Czarnikow Sugar Limited for providinginformation and insights on trading on the sugar market.

References

Goldstein, M., 1990. Influence and belief adjustment. In: Smith, J.Q., Oliver, R.M. (Eds.), Influence Diagrams, Belief Nets and Decision Analysis.Wiley, New York, pp. 143–174.

Goldstein, M., 1997. Prior inferences for posterior judgements. In: Dalla Chiara, M.L., Doets, K., Mundici, D. (Eds.), Structures and Norms inScience. Kluwer Academic, Dordrecht, pp. 55–71.

Goldstein, M., 1999. Bayes linear analysis. In: Kotz, S., Read, C., Banks, D.L. (Eds.), Encyclopedia of Statistical Sciences: Update volume 3, Wiley,New York, pp. 29–34.

Goldstein, M., O’Hagan, A., 1996. Bayes linear sufficiency and systems of expert posterior assessments. J. Roy. of Statist. Soc. B 58, 301–316.

prior viability assessment for bayesian analysis

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