  Computing the marginal likelihood or evidence is one of the core challenges
in Bayesian analysis. While there are many established methods for estimating
this quantity, they predominantly rely on using a large number of posterior
samples obtained from a Markov Chain Monte Carlo (MCMC) algorithm. As the
dimension of the parameter space increases, however, many of these methods
become prohibitively slow and potentially inaccurate. In this paper, we propose
a novel method in which we use the MCMC samples to learn a high probability
partition of the parameter space and then form a deterministic approximation
over each of these partition sets. This two-step procedure, which constitutes
both a probabilistic and a deterministic component, is termed a Hybrid
approximation to the marginal likelihood. We demonstrate its versatility in a
plethora of examples with varying dimension and sample size, and we also
highlight the Hybrid approximation's effectiveness in situations where there is
either a limited number or only approximate MCMC samples available.
