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In latent variable models the parameter estimation can be implemented byusing the joint or the marginal likelihood, based on independence orconditional independence assumptions. The same dilemma occurs within theBayesian framework with respect to the estimation of the Bayesian marginal (orintegrated) likelihood, which is the main tool for model comparison andaveraging. In most cases, the Bayesian marginal likelihood is a highdimensional integral that cannot be computed analytically and a plethora ofmethods based on Monte Carlo integration (MCI) are used for its estimation. Inthis work, it is shown that the joint MCI approach makes subtle use of theproperties of the adopted model, leading to increased error and bias in finitesettings. The sources and the components of the error associated withestimators under the two approaches are identified here and provided in exactforms. Additionally, the effect of the sample covariation on the Monte Carloestimators is examined. In particular, even under independence assumptions thesample covariance will be close to (but not exactly) zero which surprisinglyhas a severe effect on the estimated values and their variability. To addressthis problem, an index of the sample's divergence from independence isintroduced as a multivariate extension of covariance. The implicationsaddressed here are important in the majority of practical problems appearing inBayesian inference of multi-parameter models with analogous structures.

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