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I have a rather non-normal marginalized posterior for some parameters, resulting from a Bayesian MCMC. Example:

enter image description here

I know that the actual distribution is what truly represents the parameter, but I need to report a point estimate. In such a case, I believe the mode would be better than the mean or median, but I'm not sure how to justify this other than saying "the mode looks like a more reasonable point estimate for the parameter".

Is there a more "statistical" justification for selecting either point estimate?

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    $\begingroup$ Why is the MAP not equal to the posterior mode? $\endgroup$ – jbowman Jan 11 at 20:01
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    $\begingroup$ Because this is the marginalized posterior for a single parameter, and the model has 6 parameters. I obtain the MAP as the mode of the posterior distribution, not for each particular marginalized posterior. I though this is how it should be done. Is it not? $\endgroup$ – Gabriel Jan 11 at 20:08
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    $\begingroup$ A Bayes estimator can only be justified as minimising a loss function, which itself must reflect the reason for producing a point estimate rather than a full posterior distribution. $\endgroup$ – Xi'an Jan 12 at 10:43
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    $\begingroup$ I think @Xi'an is referring to the fact that any (standard) Bayesian inference through a probabilistic model is really only a specific type of loss-minimizing exercise (namely the negative log-likelihood as regularized with your prior belief). In fact, one does not even need a log-likelihood to perform Bayesian inference, any (integrable) loss function will do. While the posteriors resulting from general loss functions have been used for a while (usually named Pseudo- or Gibbs-posteriors), they do actually define coherent belief updates, see here: arxiv.org/abs/1306.6430 $\endgroup$ – Jeremias K Jan 13 at 14:17
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    $\begingroup$ Thanks for the pointer to this paper, @Jeremias K. I was not thinking of this most interesting extension, but it is quite relevant. My simpler point is that a Bayes estimator does not exist per se, since the goal of a Bayesian analysis is to return a posterior distribution. If seeking a single-value summary of this distribution, some rule need be imposed, i.e., some comparison between uni-dimensional summaries must be available, which equates for me with the notion of a loss function (also found in game theory) that is minimised across the posterior distribution as, e.g., the quadratic loss. $\endgroup$ – Xi'an Jan 13 at 15:25

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