8,650 research outputs found
Bayesian comparison of latent variable models: Conditional vs marginal likelihoods
Typical Bayesian methods for models with latent variables (or random effects)
involve directly sampling the latent variables along with the model parameters.
In high-level software code for model definitions (using, e.g., BUGS, JAGS,
Stan), the likelihood is therefore specified as conditional on the latent
variables. This can lead researchers to perform model comparisons via
conditional likelihoods, where the latent variables are considered model
parameters. In other settings, however, typical model comparisons involve
marginal likelihoods where the latent variables are integrated out. This
distinction is often overlooked despite the fact that it can have a large
impact on the comparisons of interest. In this paper, we clarify and illustrate
these issues, focusing on the comparison of conditional and marginal Deviance
Information Criteria (DICs) and Watanabe-Akaike Information Criteria (WAICs) in
psychometric modeling. The conditional/marginal distinction corresponds to
whether the model should be predictive for the clusters that are in the data or
for new clusters (where "clusters" typically correspond to higher-level units
like people or schools). Correspondingly, we show that marginal WAIC
corresponds to leave-one-cluster out (LOcO) cross-validation, whereas
conditional WAIC corresponds to leave-one-unit out (LOuO). These results lead
to recommendations on the general application of the criteria to models with
latent variables.Comment: Manuscript in press at Psychometrika; 31 pages, 8 figure
Marginal Likelihood Estimation with the Cross-Entropy Method
We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications the proposed CE method compares favorably to existing estimators
Thermodynamic assessment of probability distribution divergencies and Bayesian model comparison
Within path sampling framework, we show that probability distribution
divergences, such as the Chernoff information, can be estimated via
thermodynamic integration. The Boltzmann-Gibbs distribution pertaining to
different Hamiltonians is implemented to derive tempered transitions along the
path, linking the distributions of interest at the endpoints. Under this
perspective, a geometric approach is feasible, which prompts intuition and
facilitates tuning the error sources. Additionally, there are direct
applications in Bayesian model evaluation. Existing marginal likelihood and
Bayes factor estimators are reviewed here along with their stepping-stone
sampling analogues. New estimators are presented and the use of compound paths
is introduced
Accelerating delayed-acceptance Markov chain Monte Carlo algorithms
Delayed-acceptance Markov chain Monte Carlo (DA-MCMC) samples from a
probability distribution via a two-stages version of the Metropolis-Hastings
algorithm, by combining the target distribution with a "surrogate" (i.e. an
approximate and computationally cheaper version) of said distribution. DA-MCMC
accelerates MCMC sampling in complex applications, while still targeting the
exact distribution. We design a computationally faster, albeit approximate,
DA-MCMC algorithm. We consider parameter inference in a Bayesian setting where
a surrogate likelihood function is introduced in the delayed-acceptance scheme.
When the evaluation of the likelihood function is computationally intensive,
our scheme produces a 2-4 times speed-up, compared to standard DA-MCMC.
However, the acceleration is highly problem dependent. Inference results for
the standard delayed-acceptance algorithm and our approximated version are
similar, indicating that our algorithm can return reliable Bayesian inference.
As a computationally intensive case study, we introduce a novel stochastic
differential equation model for protein folding data.Comment: 40 pages, 21 figures, 10 table
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