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The marginal likelihood or evidence in Bayesian statistics contains anintrinsic penalty for larger model sizes and is a fundamental quantity inBayesian model comparison. Over the past two decades, there has been steadilyincreasing activity to understand the nature of this penalty in singularstatistical models, building on pioneering work by Sumio Watanabe. Unlikeregular models where the Bayesian information criterion (BIC) encapsulates afirst-order expansion of the logarithm of the marginal likelihood, parametercounting gets trickier in singular models where a quantity called the real logcanonical threshold (RLCT) summarizes the effective model dimensionality. Inthis article, we offer a probabilistic treatment to recover non-asymptoticversions of established evidence bounds as well as prove a new result based onthe Gibbs variational inequality. In particular, we show that mean-fieldvariational inference correctly recovers the RLCT for any singular model in itscanonical or normal form. We additionally exhibit sharpness of our bound byanalyzing the dynamics of a general purpose coordinate ascent algorithm (CAVI)popularly employed in variational inference.

Evidence bounds in singular models: probabilistic and variational perspectives