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.
Highlights
Variational inference (VI) (Jordan et al 1999; Wainwright et al 2008) is a powerful method to approximate intractable integrals
We demonstrate the utility of Monte Carlo co-ordinate ascent VI (MC-co-ordinate ascent VI (CAVI)) in a statistical model proposed in the field of metabolomics by Astle et al (2012), and used in NMR (Nuclear Magnetic Resonance) data analysis
The Markov chain Monte Carlo (MCMC) step of MC-CAVI is necessary to deal with parameters for which variational inference (VI) approximation distributions are difficult or impossible to derive, for example due to the impossibility to derive closed-form expression for the normalising constant
Summary
Variational inference (VI) (Jordan et al 1999; Wainwright et al 2008) is a powerful method to approximate intractable integrals. Gradient-based approaches, which can potentially scale up to large data—alluding here to recent Stochastic-Gradient-type methods—can be an effective alternative for ELBO optimisation Such algorithms have their own challenges, e.g. in the case of reparameterization Variational Bayes (VB). This is the case, e.g., for Stochastic VI (Hoffman et al 2013) and Black-Box. VI (BBVI) (Ranganath et al 2014). (2016) apply Mean-Field Variational Bayes (VB) for Generalised Linear Mixed Models, and use Monte Carlo for the approximation of analytically intractable required expectations under the variational densities; several references for related works are given in the above papers. MC-CAVI uses the Monte Carlo principle for the approximation of the difficult-to-compute conditional expectations, E−i [log p(zi− , zi , zi+ , x)], within CAVI. E.g., Forbes and Fort (2007) employ an MCMC procedure in the context of a Variational EM (VEM), to obtain estimates of the normalizing constant for Markov Random Fields—they provide asymptotic results for the correctness of the complete algorithm; Tran et al. MC-CAVI algorithm
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