To do parameter inference in an inverse problem, I use a Metroplois Hastings MCMC to sample the posterior pdf of the model parameters $\theta$ of the forward model $f(\theta)$. Where the forward model $f(\cdot)$ maps a vector $\theta$ in the parameter space $X$ to a vector $\nu$ in the space of observable features $Y$.

When sampling the pdf $p(\theta|\nu_\text{obs})$ using the MCMC, where $\nu_\text{obs}$ is a noisy observation in $Y$, I get a multivariate and unimodal pdf. Though the marginalized pdf of $p(\theta|\nu_\text{obs})$ for some of the elements of $\theta$ are highly asymmetric.

How do I best summarize such a posterior? For symmetric posteriors I would usually use the median an the 16th and 84th percentiles. For univariate asymmetric pdf I usually use the highest density interval (HDI) and the mode.

I know that there are packages to calculate the HDI of bivariate distributions, but I am not aware of any for more dimensions.


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