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I believe MCMC could be utilized to estimate the MAP. At least there is an option in packages like PyMC.

I just started reading about Bayesian Optimization, but the first thing that hit me was that it incrementally samples points from a function, much like MCMC.

Even if the nature of the function varies -- BO: black box and expensive versus MCMC: known distribution. I can image that BO can as well be utilized for a known distribution (maybe this where I am not clear yet).

My question is: how do BO and MCMC compare when it comes to MAP estimation, and in sampling efficiency in general? Specifically, is BO more efficient for MAP estimation than MCMC?

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  • $\begingroup$ This question seems much clearer, so I've voted to reopen. The question will need to accumulate a certain number of re-open votes for it to be opened again. Since my previous answer does not address the question reflected in your edit, I've deleted my answer. $\endgroup$ – Sycorax Aug 6 '18 at 15:05
  • $\begingroup$ This is about ABC instead of MCMC, but it still seems like it's relevant. "Efficient acquisition rules for model-based approximate Bayesian computation": "many ABC algorithms require a large number of simulations, which can be costly. To reduce the computational cost, Bayesian optimisation (BO) and surrogate models such as Gaussian processes have been proposed. Bayesian optimisation enables one to intelligently decide where to evaluate the model next but common BO strategies are not designed for the goal of estimating the posterior distribution. Our paper addresses this gap in the literature." $\endgroup$ – Sycorax Aug 10 '18 at 16:01
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MCMC and BO solve different problems. The purpose of MCMC is to sample from a given probability distribution, whereas the purpose of BO is to minimize a (possibly stochastic) black box function.

BO proceeds by sequentially choosing points at which to evaluate the objective function. It maintains a probabilistic model of the objective function, and uses it to choose the next point according to an 'acquisition function'. The acquisition function strikes a balance between exploring new regions to reduce uncertainty about the objective function, and exploiting known structure to improve its value.

The query points selected by BO are not samples from a user-specified probability distribution. Rather, they're chosen in a way that hopefully lets us find the minimum of the objective function quickly. Therefore, BO is not suitable when the goal is to sample from a given distribution, and is not comparable to MCMC in this context.

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  • $\begingroup$ Right. Though I am wondering how BO would perform if we use the "user-specified" distribution as if it was a black box. And assuming we are mainly interested in the MAP, which MCMC can also approximate. Can BO explore the space more efficiently if we have a non-convex function? Thanks. $\endgroup$ – DED Aug 5 '18 at 4:21
  • $\begingroup$ So, are you asking if BO is more efficient for MAP estimation than MCMC? If so, I'll edit this answer to address this, but please edit the question to clarify, as it currently reads that you're asking about sampling and/or integration, not optimization. $\endgroup$ – user20160 Aug 5 '18 at 4:51
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    $\begingroup$ Yes that’s what I meant. I phrased a generic question because I thought that the main difference is how both sample space. I know that MCMC isn’t designed for MAP, but it should provide an approximate, at least in packages like pymc. $\endgroup$ – DED Aug 5 '18 at 14:04
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    $\begingroup$ I have edited my question for added clarity. $\endgroup$ – DED Aug 6 '18 at 15:08

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