# What is the "empirical distribution" in the context of Bayesian inference?

A colleague of mine was using the functions bayesglm() and sim() from the arm-package in R to fit a Bayesian logistic regression model. I found little information on the two functions in the documentation. However, there is a short paper covering the bayesglm-function where it is stated:

The bayesglm function represents a kind of short cut of the Bayesian approach to inference. Typically, the posterior is not used directly for making inferences. Instead, an empirical distribution is constructed based on draws from the posterior and that empirical distribution is what informs the inference(s). With the bayesglm we get a distribution of ’simulates’ which are used in place of an actual empirical distribution.

https://it.unt.edu/sites/default/files/bayesglm_jds_jan2011.pdf

I have not seen this usage of the term "empirical distribution" in this context before. I guess it has to do with the approximate EM-algorithm, which is used in bayesglm instead of MCMC methods, but I am not sure what it precisely means.

Also, I have troubles to make sense out of the statement "typically, the posterior is not used directly for making inferences". Does it refer to the fact that one typically uses either samples from the posterior or an approximation, when no closed form solution for the posterior exists?

A similar question is bayesglm (arm) versus MCMCpack, but the only answer there does not contain information to answer this new question.

• I believe they mean empirical distribution = samples from the posterior (Using MCMC) Apr 8, 2021 at 13:43
• @seanv507 thanks a lot for your comment - and sorry for my slow response. I think you are correct. However, following sentence in the document is confusing me: "To make truly Bayesian inferences about our coefficients, we need to do the extra step of creating the empirical distribution(s) mentioned above, [using] the MCMCregress function in the package MCMCpack" (p.9+10). Apparently, they make a distinction between the "simulates" created with bayesglm and sim, and the posterior samples drawn with MCMC methods. Apr 8, 2021 at 14:47
• this seems the documentation stat.columbia.edu/~gelman/research/published/priors11.pdf of the method Apr 8, 2021 at 14:58