Transcript Priors - Statistics

Priors
Trevor Sweeting
Department of Statistical Science
University College London
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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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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 q is
Posterior density  Prior density x Likelihood function
 posterior probabilities, moments, marginal densities,
expected losses, predictive densities ...
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Bayesian inference
 The predictive density of Y given X is


 Where are we?...
 Philosophical basis
 Practical implementation
 Prior construction ...
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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
Subjective
Elicited prior
Mixed
Nonsubjective Default prior
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Fully
Frequentist
Performance
Penalty fn
Dual verification
Performance
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Examples
 Four examples
 All taken from Applied Statistics, 52 (2003)




Competing risks
Image analysis
Diagnostic testing
Geostatistical modelling
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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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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, f)
 k is the number of texture components, C are texture labels,
f are parameters associated with the distribution of pixel
intensities
 Highly structured prior for (k, C, f)
 Markov random field for C, truncated conjugate priors for f
 Hyperparameters set in context e.g. to ‘encourage relatively
few textures’
 Implementation: via Markov chain Monte Carlo
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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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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 zeromean 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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Table for examples
Fully
Bayesian
Subjective
Fully
Frequentist
Image
analysis
Competing risks Diagnostic
testing
Geostatistical
modelling
Nonsubjective
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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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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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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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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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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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WinBUGS - the Movie!
( f is the precision)
 Data: 529.0, 530.0, 532.0, 533.1, 533.4, 533.6, 533.7,
534.1, 534.8, 535.3
 Prior parameters: a = b = c = 0.001
 Relatively diffuse prior
 Results ...
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WinBUGS - the Movie!
Just another few iterations to make sure ...
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WinBUGS - the Movie!
Oops!
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WinBUGS - the Movie!
Effect of choice of c (the prior precision of m)
 c = 0.001 WinBUGS eventually gets the ‘right’ answer
 but presumably not the answer we wanted!
Marginal posterior density of m
1.40
Prior precision = 0.001
1.20
density
1.00
0.80
0.60
0.40
0.20
-300
0.00
-100
100
300
500
700
900
1100
m
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The ‘noninformative’ prior dominates the likelihood.
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WinBUGS - the Movie!
 c = 0.0002 WinBUGS gives the ‘right’ answer with the likelihood
dominating
 However, it's the ‘wrong’ answer as the true marginal posterior of m
is still dominated by the prior
density
Marginal posterior density of m
-300
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
-100
Prior precision = 0.0002
100
300
500
700
900
1100
m
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WinBUGS - the Movie!
 c = 0.00016 WinBUGS again gives the ‘right’ answer with the
likelihood dominating
 But it's still the ‘wrong’ answer
density
Marginal posterior density of m
-300
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
-100
Prior precision = 0.00016
100
300
500
700
900
1100
m
 The true marginal posterior distribution of m is bimodal
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WinBUGS - the Movie!
 c = 0.00010 WinBUGS gives the right answer
 ... and presumably the one we wanted!
density
Marginal posterior density of m
-300
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
-100
Prior precision = 0.0001
100
300
500
700
900
1100
m
Care needed in the choice of prior parameters in
diffuse but proper priors
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Normal regression
( f is the precision)
 Conjugate prior:
 Limit as
is

 Jeffreys' prior
 Here
gives exact matching in both posterior
and predictive distributions
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Normal regression
 Data: n = 25, R = residual sum of squares = 2.1
1.
2.
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Normal regression
 Prediction. Let Y be a future observation and let
denote the ‘usual’ predictive pivotal quantity. Then
1.
2.
Prediction less sensitive to prior than estimation
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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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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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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 a!
 If there does exist a matching prior that is free from a 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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Relative entropy loss
 The ‘reference prior’ (Bernardo, 1979) maximises the
Shannon mutual information between q 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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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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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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Relative entropy loss
 Define
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 p under repeated use or under alternative subjective
priors t
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Relative entropy loss
 Now define the predictive relative entropy loss
(PREL)
 where J is Jeffreys’ prior
 Studying the behaviour of the regret
over t in
sets of constant 'predictive information' is equivalent to
studying the behaviour of the PREL
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Relative entropy loss
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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
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the (asymptotic) predictive loss
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Relative entropy loss
 More generally define

represents the asymptotically worst-case loss
 Investigate its behaviour
 Let
 The prior
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is minimax if
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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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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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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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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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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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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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Wrap-up
 And finally
Have a great conference!
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