How do I compute the standard error of the posterior mode? How do I compute the standard error of the posterior mode? I've tried checking various sources like Google and Bing, but I haven't been able to answer my question.
 A: As mentioned in the comments, the standard error is a concept in frequentist statistics. If you want to report some measure of uncertainty, I would suggest that you construct a Credible Interval.
Now, credible intervals are not unique as you can choose to "cut" the posterior distribution in many different ways and get different intervals every time you do it. Hence, given that you are interested on the posterior mode and assuming that your posterior distribution is unimodal, I recommend that you take the highest posterior density interval (HPDI) which is is narrowest credible interval and includes the mode.
A: The answer by Vas is correct. Credible intervals are the most Bayesian way. However I understand there is sometimes a need to present a simple index of uncertainty.
My first thought is to estimate the fisher information / Hessian (making a variational approximation). The square root of the diagonals of this should give a 'standard error'-like quantity, in the same units as the parameter of interest. For posterior $q(\theta)$,
$$
H_{ij} =\left[ \frac{\partial^2 \ \mathrm{log}(q)}{\partial\theta_{ij}} \right]
$$
$$
std. err. \approx \sqrt{H_{ii}}
$$
Most optimisation tools give Hessian e.g. fmincon.
Note the caveats with this approach - assumes unimodal, gaussian densities. And isn't Bayesian.
The benefits - works with multivariate posteriors, and you can see axes of correlations between parameters as eigenvectors of $H$.
Here is a more formal answer:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5608466/
