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Title: Challenges in Computing and Optimizing Upper Bounds of Marginal Likelihood based on Chi-Square Divergences

Paper content:
Variational inference based on chi-square divergence minimization (CHIVI) provides a way to approximate a model's posterior while obtaining an upper bound on the marginal likelihood.
However, in practice CHIVI relies on Monte Carlo (MC) estimates of an upper bound objective that at modest sample sizes are not guaranteed to be true bounds on the marginal likelihood.
This paper provides an empirical study of CHIVI performance on a series of synthetic inference tasks.
We show that CHIVI is far more sensitive to initialization than classic VI based on KL minimization, often needs a very large number of samples (over a million), and may not be a reliable upper bound.
We also suggest possible ways to detect and alleviate some of these pathologies, including diagnostic bounds and initialization strategies.
Estimating the marginal likelihood in probabilistic models is the holy grail of Bayesian inference.
Marginal likelihoods allow us to compute the posterior probability of model parameters or perform Bayesian model selection (Bishop et al., 1995) .
While exact computation of the marginal is not tractable for most models, variational inference (VI) (Jordan et al., 1999 ) offers a promising and scalable approximation.
VI suggests choosing a simple family of approximate distributions q and then optimizing the parameters of q to minimize its divergence from the true (intractable) posterior.
The canonical choice is the KL divergence, where minimizing corresponds to tightening a lower bound on the marginal likelihood.
Recently, (Dieng et al., 2017a) showed that minimizing a χ 2 divergence leads to a chi-divergence upper bound ("CUBO").
Practitioners often wish to combine upper and lower bound estimates to "sandwich" the model evidence in a narrow range for later decision making, so the CUBO's flexible applicability to all latent variable models is appealing.
However, both the estimation of the upper bound and computing its gradient for minimization require Monte Carlo estimators to approximate tough integrals.
These estimators may have large variance even at modest number of samples.
A natural question is then how reliable CUBO minimization is in practice.
In this paper, we provide empirical evidence that CUBO optimization is often tricky, and the bound itself ends up being too loose even Figure 1: Minimizing χ 2 divergence using MC gradient estimates via the reparametrization trick can be challenging even with simple univariate Gaussian distributions.
Each column shows results under a different number of MC samples.
The last column compares ELBO and CUBO traces for S = 10 4 ; diamonds correspond to sanity-check estimator from Eq. (2).
Top row : variational parameter traces with fixed true variance but changing starting mean locations.
Bottom row: same, but with fixed true mean and changing start variance values.
using hundreds of samples.
Our contributions include:
i) evaluation of the CUBO in two simple scenarios, and comparison to other bounds to gauge its utility;
ii) empirical analysis of CUBO optimization in both scenarios, in terms of convergence rate and sensitivity to the number of samples;
iii) review of alternative upper bounds and best practices for diagnosing and testing new bounds.

assistant: text: An empirical study of variational inference based on chi-square divergence minimization, showing that minimizing the CUBO is trickier than maximizing the ELBO