Fast computation of the deviance information criterion for latent variable models

Abstract

The deviance information criterion (DIC) has been widely used for Bayesian model comparison. However, recent studies have cautioned against the use of certain variants of the DIC for comparing latent variable models. For example, it has been argued that the conditional DIC-based on the conditional likelihood obtained by conditioning on the latent variables-is sensitive to transformations of latent variables and distributions. Further, in a Monte Carlo study that compares various Poisson models, the conditional DIC almost always prefers an incorrect model. In contrast, the observed-data DIC-calculated using the observed-data likelihood obtained by integrating out the latent variables-seems to perform well. It is also the case that the conditional DIC based on the maximum a posteriori (MAP) estimate might not even exist, whereas the observed-data DIC does not suffer from this problem. In view of these considerations, fast algorithms for computing the observeddata DIC for a variety of high-dimensional latent variable models are developed. Through three empirical applications it is demonstrated that the observed-data DICs have much smaller numerical standard errors compared to the conditional DICs. The corresponding Matlab code is available upon request.

Comparing hierarchical models via the marginalized deviance information criterion

“…However, this approximation holds only for large samples and requires of non‐parametric density approximation to the posterior p ( μ i | y i , Ψ ) when the latent variables vector is not Gaussian. Some methods for fast computation of mDIC are advocated in Chan and Grant for several cases where the marginalized likelihood is available analytically. It is shown empirically that the variability of cDIC is importantly larger compared with mDIC, as expected, since the former depends on the latent unknown variables whereas the marginalized criterion integrates them out.…”

Trend and Cycle Shocks in Bayesian Unobserved Components Models for UK Productivity

“…Furthermore, the UC models are always likelier than enforcing an H-P …lter or a deterministic trend on the data. 11 Second, allowing for the large, non-informative priors for the trend and cycle component shocks results in the posterior estimates of the trend shocks being relatively large compared to the cycle shocks (for similar results with US GDP, see Grant and Chan (2017a)).…”

“…In case 6, the shock originates from the cycle equation, and this has a negative correlation with the trend shock. In equation (11), if, at time t, there is a shock to t of size x t , which causes the cycle to change by ax t , and it also causes t to change by z t and consequently the trend changes by bz t (which has the opposite sign to ax t , given the negative correlation assumed). Consequently, as in case 5, y t = ax t + bz t , which is smaller in absolute value than ax t .…”

“…Typically, maximum likelihood methods can lead to di¢ culties in …nding solutions for the models when the number of parameters to be estimated is large relative to the number of observations (as is often the case in correlated UC models). In doing so, we use recent techniques on Bayesian methods for correlated UC models developed by Chan and Eisenstat (2015), Chan and Grant (2016) and Grant and Chan (2017a) and (2017b). Furthermore, we also present some extensions and modi…cations to the latter modelling framework by allowing for di¤erent speci…cations for the trend component of the models.…”

Business cycle dynamics after the Great Recession

“…This information criterion is relatively easy to compute when model estimation is based on Bayesian Markov Chain Monte Carlo (MCMC) techniques, like Gibbs Sampling. Nevertheless, the literature shows that DIC estimation is all the more accurate that the number of parameters in the conditioning set of the log-likelihood is limited (Chan and Grant 2016). Here, we compute the conditional log-likelihood from the Kalman filter step of the Gibbs Sampler.…”