I was wondering if you wanted to compute the MAP estimate of an unknown posterior distribution, is there a non-sampling based method that would suffice? As in, if you don’t need to know anything more than the posterior mode, what would be the method to approximate this?

Two related questions Why is MCMC needed when estimating a parameter using MAP and Are MCMC based methods appropriate when Maximum a-posteriori estimation is available? but they don't quite answer my question.

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    $\begingroup$ Usually the problem with Bayesian computation is calculating the normalizing constant; in many simple cases you don't need this to simply find the MAP estimate. But what exactly do you mean by "an unknown posterior distribution"? $\endgroup$ – Maurits M May 14 '19 at 7:45
  • $\begingroup$ As in, a posterior distribution you don't know the probability density (shape) of. $\endgroup$ – questionmark May 14 '19 at 10:45
  • $\begingroup$ Do you know anything about the posterior? Usually you know the posterior up to a normalizing constant (i.e. you know the prior and likelihood and can compute the product, but not the integral of the product). If you know that much, you can maximize it using normal optimization methods (gradient based or otherwise) to find the MAP. You don't need any MCMC for that. I think this is well described in the first post you link to. $\endgroup$ – Maurits M May 14 '19 at 12:10

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