Deviance information criterion

The deviance information criterion (DIC) is a hierarchical modeling generalization of the AIC (Akaike information criterion) and BIC (Bayesian information criterion, also known as the Schwarz criterion). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. Like AIC and BIC it is an asymptotic approximation as the sample size becomes large. It is only valid when the posterior distribution is approximately multivariate normal.

Define the deviance as $D(\theta )=-2\log(p(y|\theta ))+C\,$, where $y\,$ are the data, $\theta \,$ are the unknown parameters of the model and $p(y|\theta )\,$ is the likelihood function. $C\,$ is a constant that cancels out in all calculations that compare different models, and which therefore does not need to be known.

The expectation ${\bar {D}}=\mathbf {E} ^{\theta }[D(\theta )]$ is a measure of how well the model fits the data; the larger this is, the worse the fit.

The effective number of parameters of the model is computed as $p_{D}={\bar {D}}-D({\bar {\theta }})$, where ${\bar {\theta }}$ is the expectation of $\theta \,$. The larger this is, the easier it is for the model to fit the data.

The deviance information criterion is calculated as

${\mathit {DIC}}=p_{D}+{\bar {D}}.$

The idea is that models with smaller DIC should be preferred to models with larger DIC. Models are penalized both by the value of ${\bar {D}}$, which favors a good fit, but also (in common with AIC and BIC) by the effective number of parameters $p_{D}\,$. Since ${\bar {D}}$ will decrease as the number of parameters in a model increases, the $p_{D}\,$ term compensates for this effect by favoring models with a smaller number of parameters.

The advantage of DIC over other criteria in the case of Bayesian model selection is that the DIC is easily calculated from the samples generated by a Markov chain Monte Carlo simulation. AIC and BIC require calculating the likelihood at its maximum over $\theta \,$, which is not readily available from the MCMC simulation. But to calculate DIC, simply compute ${\bar {D}}$ as the average of $D(\theta )\,$ over the samples of $\theta \,$, and $D({\bar {\theta }})$ as the value of $D\,$ evaluated at the average of the samples of $\theta \,$. Then the DIC follows directly from these approximations. Claeskens and Hjort (2008, Ch. 3.5) show that the DIC is large-sample equivalent to the natural model-robust version of the AIC.

In the derivation of DIC, it assumed that the specified parametric family of probability distributions that generate future observations encompasses the true model. This assumption does not always hold, and it is desirable to consider model assessment procedures in that scenario. Also, the observed data are used both to construct the posterior distribution and to evaluate the estimated models. Therefore, DIC tends to select over-fitted models. Recently, these issues are resolved by Ando (2007), Bayesian predictive information criterion, BPIC.

To avoid the over-fitting problems of DIC, Ando (2012) developed Bayesian model selection criteria from a predictive view point. The criterion is calculated as

${\mathit {IC}}=\mathbf {E} ^{\theta }[-2\log(p(y|\theta ))]+2p_{D}.$

See also

• Akaike information criterion (AIC)
• Bayesian information criterion (BIC)
• Bayesian predictive information criterion (BPIC)
• Focused information criterion (FIC)
• Kullback–Leibler divergence
• Jensen–Shannon divergence

References

• Claeskens, G, and Hjort, N.L. (2008). Model Selection and Model Averaging, Cambridge. Section 3.5.
• Gelman, Andrew; Carlin, John B.; Stern, Hal. S.; Rubin, Donald B. (2004). Bayesian Data Analysis (2nd ed.). Boca Raton, Florida: Chapman & Hall/CRC. pp. 182–184. ISBN 1-58488-388-X. MR 2027492. LCC QA279.5.B386 2004.
• Spiegelhalter, David J.; Best, Nicola G.; Carlin, Bradley P.; van der Linde, Angelika (2002). "Bayesian measures of model complexity and fit (with discussion)". Journal of the Royal Statistical Society, Series B (Statistical Methodology). 64 (4): 583–639. doi:10.1111/1467-9868.00353. JSTOR 3088806. MR 1979380. {{cite journal}}: Unknown parameter |month= ignored (help)
• Ando, Tomohiro (2007). "Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models". Biometrika. 94 (2): 443–458. doi:10.1093/biomet/asm017.
• Ando, Tomohiro (2012). "Predictive Bayesian model selection". American Journal of Mathematical and Management Sciences.