# How to interpret posterior distribution plots for multiple priors? [closed]

Let us consider the following Bayesian model:

• $f(x|\mu, \theta)$ the observation model,
• $\pi(\mu, \theta)$ the joint prior distribution,
• $\pi(\mu, \theta|x)$ the posterior distribution.

It is possible to obtain the posterior distribution with a MCMC sampler. Then the posterior distributions can be plotted. Using Python pymc3 plot_posterior function for example, two pdf figures would be obtained: one for $\mu$, one for $\theta$. It seems these figures are fairly standard (I saw them too in Matlab documentation), so I doubt this is a pymc3 specific feature.

How do these figures relate to $\pi(\mu, \theta|x)$?

I would say the model you present has one prior for multiple parameters (in this case two: $\mu$ and $\theta$). Because there are two parameters, the prior is a joint prior. Similarly, there is a single joint posterior distribution for the two parameters: $\pi(\mu,\theta|x)$. I'm guessing the software that you refer to produces plots of the two marginal posterior distributions, $$\pi(\mu|x) = \int \pi(\mu,\theta|x)\,d\theta$$ and $$\pi(\theta|x) = \int \pi(\mu,\theta|x)\,d\mu .$$ In order to plot the joint posterior distribution (I'm guessing) you would need a different command.
• I believe this answer is spot on; the only thing I can add to help answer the question is that from the MCMC sampler you should have paired samples for $\mu$ and $\theta$. You can visualise $\pi ( \mu , \theta | x )$ by doing a 2 variable density plot of these samples something like this perhaps Commented Mar 9, 2018 at 18:39