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BIC is an acronym for Bayesian Information Criterion. BIC is one method of model comparison. See also AIC

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Is it circular reasoning to compute the ELBO using MCMC?

On the other hand, in the BIC we also get an parameter value $\hat{\theta}_{ML}$ from maximizing the function we try to estimate. So is this something one can do? … And if yes, would it make sense to improve the BIC by using the MCMC estimator for the ELBO instead of the maximum log-likelihood? …
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