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RSC Priors Trevor Sweeting Department of Statistical Science University College London

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RSC Structure of talk Bayesian inference: the basics Specification of the prior Examples Subjective priors Nonsubjective priors Examples Methods of prior construction Coverage probability bias Relative entropy loss Wrap-up

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RSC Bayesian inference: the basics X – the experimental or observational data to be observed Y – the future observations to be predicted Data model (Possibly improper) prior distribution The posterior density of is Posterior density Prior density x Likelihood function posterior probabilities, moments, marginal densities, expected losses, predictive densities...

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RSC Bayesian inference The predictive density of Y given X is Where are we?... Philosophical basis Practical implementation Prior construction...

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RSC Specification of the prior Approaches vary from fully Bayesian analyses based on fully elicited subjective priors to fully frequentist analyses based on nonsubjective (‘objective’) priors Fully Bayesian Fully Frequentist SubjectiveElicited prior MixedPerformancePenalty fn NonsubjectiveDefault priorDual verificationPerformance

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RSC Examples Four examples All taken from Applied Statistics, 52 (2003) Competing risks Image analysis Diagnostic testing Geostatistical modelling

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RSC Competing risks (Basu and Sen) System failure data; cause of failure not identified n systems, R competing risks Datum for each system is (T, S, C) T is failure time, S are the possible causes of failure, C is a censoring indicator Parameters in the model are of location & scale type Use (i) informative conjugate priors Source: historical data or (ii) ‘noninformative’ priors Such that they have a ‘minimal effect’ on the analysis Implementation: via Gibbs sampling

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RSC Image analysis (Dryden, Scarr and Taylor) Segmentation of weed and crop textures Automatic identification of weeds in images of row crops Parameters are (k, C, ) k is the number of texture components, C are texture labels, are parameters associated with the distribution of pixel intensities Highly structured prior for (k, C, ) Markov random field for C, truncated conjugate priors for Hyperparameters set in context e.g. to ‘encourage relatively few textures’ Implementation: via Markov chain Monte Carlo

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RSC Diagnostic testing (Georgiadie, Johnson, Gardner and Singh) Multiple-test screening data models are unidentifiable A Bayesian analysis therefore depends critically on prior information Parameters consist of various (at least 8) joint sensitivity and specificity probabilities Independent beta priors; two informative, the rest noninformative Investigate coverage performance and sensitivities for various choices of prior Implementation: via Gibbs sampling

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RSC Geostatistical modelling (Kammann and Wand) Geostatistical mapping to study geographical variability of reproductive health outcomes (disease mapping) Geoadditive models Universal kriging model involves a stationary zero- mean stochastic process over sites leads to ‘borrowing strength’ Non-Bayesian analysis, but model could be formulated in a Bayesian way, with the mean responses at the given sites having a multivariate normal prior Implementation: residual ML and splines

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RSC Table for examples Fully Bayesian Fully Frequentist Subjective Nonsubjective Competing risks Image analysis Diagnostic testing Geostatistical modelling

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RSC Subjective priors To some extent, all the previous examples included subjective prior specification Methods of elicitation Industrial and medical contexts Scientific reporting Range of prior specifications; conduct sensitivity analyses

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RSC Subjective priors Psychological research: should take account of when devising methods for prior elicitation Construction of questions Anchors Probability assessment by frequency Availability; inverse expertise effect Priors are often ‘too narrow’ Experimental Psychology, Behavioural Decision Making, Management Science, Cognitive Psychology

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RSC Nonsubjective priors Nonsubjective (‘objective’) priors: why? Sensible default priors for non-experts (and experts!) Recognise basis often weak Possible nasty surprises! Reference priors for regulatory bodies Clinical trials, industrial standards, official statistics Safe default priors for high-dimensional problems Priors more difficult to specify and possibly more severe effect

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RSC Nonsubjective priors Some general problems Improper priors Improper posteriors E.g. Hierarchical models Marginalisation and sampling theory paradoxes Dutch books Inconsistency Posterior doesn’t concentrate around true value asymptotically Inadmissibility of Bayes decision rules/estimators

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RSC Nonsubjective priors Proper ‘diffuse’ priors Near-impropriety of posterior Unintended large impact on posterior Example to follow... Arbitrary choice of hyperparameters Non-objectivity Lack of invariance Egg on face... Two examples...

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RSC WinBUGS - the Movie! Data: 529.0, 530.0, 532.0, 533.1, 533.4, 533.6, 533.7, 534.1, 534.8, Prior parameters: a = b = c = Relatively diffuse prior Results... ( is the precision)

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RSC WinBUGS - the Movie! Just another few iterations to make sure...

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RSC WinBUGS - the Movie! Oops!

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RSC WinBUGS - the Movie! Effect of choice of c (the prior precision of ) c = WinBUGS eventually gets the ‘right’ answer but presumably not the answer we wanted! The ‘noninformative’ prior dominates the likelihood.

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RSC WinBUGS - the Movie! c = WinBUGS gives the ‘right’ answer with the likelihood dominating However, it's the ‘wrong’ answer as the true marginal posterior of is still dominated by the prior

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RSC WinBUGS - the Movie! c = WinBUGS again gives the ‘right’ answer with the likelihood dominating But it's still the ‘wrong’ answer The true marginal posterior distribution of is bimodal

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RSC WinBUGS - the Movie! c = WinBUGS gives the right answer ... and presumably the one we wanted! Care needed in the choice of prior parameters in diffuse but proper priors

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RSC Normal regression Conjugate prior: Limit as is Jeffreys' prior Here gives exact matching in both posterior and predictive distributions ( is the precision)

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RSC Normal regression Data: n = 25, R = residual sum of squares =

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RSC Normal regression Prediction. Let Y be a future observation and let denote the ‘usual’ predictive pivotal quantity. Then Prediction less sensitive to prior than estimation

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RSC Methods of prior construction Limits of proper priors Uniform priors/choice of scale Data-translated likelihood Constant asymptotic precision Canonical parameterisation Coverage Probability Bias Decision-theoretic

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RSC Coverage probability bias Sometimes investigated in papers via simulation (cf. the diagnostic testing example) Parametric CPB When do Bayesian credible intervals have the correct frequentist coverage? In regular one-parameter problems, ‘matching’ is asymptotically achieved by Jeffreys' prior (Welch and Peers, 1963) In multiparameter families cannot in general achieve matching for all marginals using the same prior Usually contravenes the likelihood principle (see Sweeting, 2001 for a discussion) Avoid infinite confidence sets! (e.g. ratios of parameters)

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RSC Coverage probability bias Predictive CPB When do Bayesian predictive intervals have the correct frequentist coverage? In regular one-parameter problems, there exists a unique prior for which there's no asymptotic CPB... ... but in general this depends on the probability level ! If there does exist a matching prior that is free from then it is Jeffreys' prior (Datta, Mukerjee, Ghosh and Sweeting, 2000) In the multiparameter case, if there exists a matching prior then it is usually not Jeffreys' prior

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RSC Relative entropy loss The ‘reference prior’ (Bernardo, 1979) maximises the Shannon mutual information between and X Maximises the ‘distance’ between the prior and posterior; minimal effect of the prior Also arises as an asymptotically minimax solution under relative entropy loss (Clarke and Barron, 1994, Barron, 1998)

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RSC Relative entropy loss Define the prior-predictive regret Minimax/reference prior solution for the full parameter is usually Jeffreys' prior Bernardo argues that when nuisance parameters are present the reference prior should depend on which parameter(s) are considered to be of primary interest

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RSC Relative entropy loss A predictive relative entropy approach Geisser (1979) suggested a predictive information criterion introduced by Aitchison (1975) Standard argument for using log q(Y) as an operational/default utility function for q as a predictive density for a future observation Y (c.f. Good, 1968)

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RSC Relative entropy loss is the expected regret under the loss function associated with using the predictive density when Y arises from Appropriate object to study for constructing objective prior distributions when we are interested in predictive performance of under repeated use or under alternative subjective priors Define

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RSC Relative entropy loss Now define the predictive relative entropy loss (PREL) where J is Jeffreys’ prior Studying the behaviour of the regret over in sets of constant 'predictive information' is equivalent to studying the behaviour of the PREL

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RSC Relative entropy loss

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RSC Relative entropy loss Under suitable regularity conditions we get Although the defined loss functions cover an infinite variety of possibilities for (a) amount of data to be observed and (b) predictions to be made, they are all approximately equivalent to provided that a sufficient amount of data is to be observed. Call the (asymptotic) predictive loss

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RSC Relative entropy loss More generally define represents the asymptotically worst-case loss Investigate its behaviour Let The prior is minimax if

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RSC Relative entropy loss Example 1. Consider the class of improper priors These all deliver constant risk, with All the priors with c nonzero are therefore inadmissible Jeffreys' prior (c = 0) is minimax

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RSC Relative entropy loss Example 2. Consider the class of improper priors These all deliver constant risk, with L attains its minimum value when a = 1, which corresponds to Jeffreys' independence prior The minimum value -½ < 0 so that Jeffreys' prior is inadmissible

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RSC Relative entropy loss Example 3. Consider again the class of improper priors These all deliver constant risk, with L attains its minimum value when a = 1, which again corresponds to Jeffreys' independence prior The drop in predictive loss increases as the square of the number q of regressors in the model

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RSC Relative entropy loss The above predictive minimax priors also give rise to minimum predictive coverage probability bias (Datta, Mukerjee, Ghosh and Sweeting, 2000) Final note: an inappropriately elicited subjective prior may lead to very high predictive risk!

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RSC Wrap-up We have reviewed some common approaches to prior construction, from full elicitation to using default recipes Need to be aware of dangers, whatever the approach As model complexity increases it becomes more difficult to make sensible prior assignments. At the same time, the effect of the prior specification can become more pronounced Important to have a sound methodology for the construction of priors in the multiparameter case Data-dependent priors may be justifiable (e.g. Box-Cox transformation model)

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RSC Wrap-up More extensive analysis of the predictive risk approach needed Developing general methods of finding exact and approximate solutions for practical implementation Investigating connections with predictive coverage probability bias Analysing dependent and non-regular problems Investigating problems involving mixed subjective/nonsubjective priors Priors for model choice or model averaging... ... another talk!

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RSC Wrap-up And finally Have a great conference!

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