Bayesian counterpart to parameter estimate precision

Question

In maximum likelihood theory it is common to summarise parameter estimates by their maxium likelihood estimate $\theta_{\mathrm{MLE}}$ and the corresponding standard error $\sigma_{\mathrm{MLE}}$ or coefficient of variation $$CV = \frac{\sigma_{\mathrm{MLE}}}{\theta_{\mathrm{MLE}}}.$$ This works since we assume that the MLE is normally distributed.

Especially, when using the $CV$ it is easy to understand the precision of the estimate independent of the scale of the parameter.

In Bayesian statistics, we get the posterior density for $\theta$ $$p(\theta | \mathcal{D}) \propto p(\mathcal{D} | \theta) p(\theta).$$ From this we can calculate the mean, the mode and whichever credible interval we want. However, I am having a hard time to find a reasonable equivalent for the kind of scale free precision estimate I can get from the $CV$ in the maximum likelihood case.

The problem here is that my posterior parameter distribution does not have an analytical form. I have it only in the form of MCMC samples.

This seems like such a standard question, but I was quite surprised that I didn't seem to find anything sensible on Google.

My idea so far is to present median and mode, as well as 68% and 95% credible intervals, to give readers a sense of comparison with the normal distribution. But I want to also be able to tell scale-free if my estimate has a good precision or not. Compared to my prior it is localized, but how would I tell if precision is good?

I feel like I might have misunderstood something fundamental here.

EDIT To clarify my question:

Assume I have two parameters in my model $\theta_1$ and $\theta_2$ and assume that $\theta_1$'s magnitude is somewhere around 0.3 and $\theta_2$'s somewhere around 13. These parameters fulfill different rolls in my (non-linear) model, so that they are both impactful. In a maximum likelihood analysis, I could present the $CV$ of these parameters which normalizes the standard deviation by the MLE estimate and therefore is scale free.

My main question is if there is a standard procedure for this in Bayesian analysis? Or do I have to come up with my own normalization?

Since I have a non-linear model, maybe it would be necessary to normalize the posterior distributions spread by the sensitivity of the parameter?

Answer 1

Given that $\theta_{\text{MLE}}$ is a point estimator, the obvious counterpart in Bayesian analysis would be the posterior mode estimator $\theta_{\text{MAP}} \equiv \arg \max_\theta f( \theta | x )$. Although this is a Bayesian estimator, you could still derive its frequentist sampling properties, including its standard error and corresponding coefficient of variation, just as you can with the MLE. Both the MLE and MAP estimators have asymptotic normal distributions under appropriate regularity conditions. In Bayesian analysis it is common to encounter problems where the posterior mode does not have a closed form solution and so you would obtain this via MCMC methods, and similarly, its frequentist properties would also be obtained via numerical methods.

Answer 2

I don't think you are missing anything - and your plan seems sensible. After Bayesian/MCMC analysis it's common to simply summarise the posterior for the parameters or the effects of interest (i.e. the MCMC chains).

The tidybayes package implements a bunch of functions to make this easy (e.g. mean_qi or mean_hdi). Jon Kruscke's book, Doing Bayesian Analysis, is helpful on this.

I found it similarly disconcerting how easy it was to explore/visualise the model posterior to explain my models... but I'm now converted. I particularly like the simplicity of making predictions from a model (e.g. tidybayes::add_fitted_samples) and summarising differences between predictions at covariate values of interest.