Monte Carlo co-ordinate ascent variational inference

Abstract

In variational inference (VI), coordinate-ascent and gradient-based approaches are two major types of algorithms for approximating difficult-to-compute probability densities. In real-world implementations of complex models, Monte Carlo methods are widely used to estimate expectations in coordinate-ascent approaches and gradients in derivative-driven ones. We discuss a Monte Carlo co-ordinate ascent VI (MC-CAVI) algorithm that makes use of Markov chain Monte Carlo (MCMC) methods in the calculation of expectations required within co-ordinate ascent VI (CAVI). We show that, under regularity conditions, an MC-CAVI recursion will get arbitrarily close to a maximiser of the evidence lower bound with any given high probability. In numerical examples, the performance of MC-CAVI algorithm is compared with that of MCMC and—as a representative of derivative-based VI methods—of Black Box VI (BBVI). We discuss and demonstrate MC-CAVI’s suitability for models with hard constraints in simulated and real examples. We compare MC-CAVI’s performance with that of MCMC in an important complex model used in nuclear magnetic resonance spectroscopy data analysis—BBVI is nearly impossible to be employed in this setting due to the hard constraints involved in the model.