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?

  • 1
    $\begingroup$ Why is the MAP not equal to the posterior mode? $\endgroup$ – jbowman Jan 11 at 20:01
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ No, you're right. Thanks for clarifying. $\endgroup$ – jbowman Jan 11 at 20:08
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.