# Bayesian measures of model complexity and fit

Bayesian measures of model complexity and fit Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the Royal Statistical Society: Series B (Statistical Methodology) Wiley

# Bayesian measures of model complexity and fit

57 pages

/lp/wiley/bayesian-measures-of-model-complexity-and-fit-P1FSsEPC0E
Publisher
Wiley
ISSN
1369-7412
eISSN
1467-9868
DOI
10.1111/1467-9868.00353
Publisher site
See Article on Publisher Site

### Abstract

Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

### Journal

Journal of the Royal Statistical Society: Series B (Statistical Methodology)Wiley

Published: Jan 1, 2002

Keywords: ; ; ; ; ; ; ; ;

### References

• Asymptotic behaviour of Bayes estimates under possibly incorrect models
Bunke, O.; Milhaud, X.
• Model Selection and Inference
Burnham, K. P.; Anderson, D. R.
• Analysis of multivariate probit models
Chib, S.; Greenberg, E.
• Empirical Bayes estimates of age‐standardised relative risks for use in disease mapping
Clayton, D. G.; Kaldor, J.
• Conditional categorical response models with application to treatment of acute myocardial infarction
Gelfand, A. E.; Ecker, M. D.; Christiansen, C.; McLaughlin, T. J.; Soumerai, S. B.
• Random‐effects models for longitudinal data using Gibbs sampling
Gilks, W. R.; Wang, C. C.; Coursaget, P.; Yvonnet, B.
• The surprise index for the multivariate normal distribution
Good, I. J.
• Counting degrees of freedom in hierarchical and other richly‐parameterised models
Hodges, J.; Sargent, D.
• Random effects models for longitudinal data
Laird, N. M.; Ware, J. H.
• Distribution of informational statistics and a criterion for model fitting (in Japanese)
Takeuchi, K.
• Technical Report
Ye, J.; Wong, W.
• Proc. 2nd Int. Symp. Information Theory
Akaike, H.
• A note on the generalized information criterion for choice of a model
Atkinson, A. C.
• Expected information as expected utility
Bernardo, J. M.
• Experts in Uncertainty
Cooke, R. M.
• Probabilistic Networks and Expert Systems
Cowell, R. G.; Dawid, A. P.; Lauritzen, S. L.; Spiegelhalter, D. J.

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations