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We consider the estimation of the marginal likelihood in Bayesian statistics, with primary emphasis on Gaussian graphical models, where the intractability of the marginal likelihood in high dimensions is a frequently researched problem. We propose a general algorithm that can be widely applied to a variety of problem settings and excels particularly when dealing with near log-concave posteriors. Our method builds upon a previously posited algorithm that uses MCMC samples to partition the parameter space and forms piecewise constant approximations over these partition sets as a means of estimating the normalizing constant. In this paper, we refine the aforementioned local approximations by taking advantage of the shape of the target distribution and leveraging an expectation propagation algorithm to approximate Gaussian integrals over rectangular polytopes. Our numerical experiments show the versatility and accuracy of the proposed estimator, even as the parameter space increases in dimension and becomes more complicated.