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 (ELBO) 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 (NMR) spectroscopy data analysis -- BBVI is nearly impossible to be employed in this setting due to the hard constraints involved in the model.
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