Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions

In latent variable models parameter estimation can be implemented by using the joint or the marginal likelihood, based on independence or conditional independence assumptions. The same dilemma occurs within the Bayesian framework with respect to the estimation of the Bayesian marginal (or integrated) likelihood, which is the main tool for model comparison and averaging. In most cases, the Bayesian marginal likelihood is a high dimensional integral that cannot be computed analytically and a plethora of methods based on Monte Carlo integration (MCI) are used for its estimation. In this work, it is shown that the joint MCI approach makes subtle use of the properties of the adopted model, leading to increased error and bias in finite settings. The sources and the components of the error associated with estimators under the two approaches are identified here and provided in exact forms. Additionally, the effect of the sample covariation on the Monte Carlo estimators is examined. In particular, even under independence assumptions the sample covariance will be close to (but not exactly) zero which surprisingly has a severe effect on the estimated values and their variability. To address this problem, an index of the sample’s divergence from independence is introduced as a multivariate extension of covariance. The implications addressed here are important in the majority of practical problems appearing in Bayesian inference of multi-parameter models with analogous structures.     

Thermodynamic Bayesian model comparison

Thermodynamics have been shown to have direct applications in Bayesian model evaluation. Within a tempered transitions scheme, the Boltzmann–Gibbs distribution pertaining to different Hamiltonians is implemented to create a path which links the distributions of interest at the endpoints. As illustrated here, an optimal temperature exists along the path which directly provides the free energy, which in this context corresponds to the marginal likelihood and/or Bayes factor. Estimators which have been developed under this framework are organised here using a unifying approach, in parallel with their stepping-stone sampling counterparts. New estimators are presented and the use of compound paths is introduced. As a byproduct, it is shown how the thermodynamic integral allows for the estimation of probability distribution divergences and measures of statistical entropy. A geometric approach is employed here to illustrate the importance of the choice of the path in terms of the corresponding estimator’s error (path-related variance), which provides a more intuitive approach in tuning the error sources.