Parameter Values From Asymmetric Probability Distributions

Question

I am performing a Multipole Decomposition Analysis on some experimental data, essentially fitting a set of experimental data with a linear combination of functions. Annoyingly these functions are not analytic nor are they orthogonal to each other. Below is an example of such a fit.

My fitted parameters are the coefficients of the functions (which are in the interval [0, 1]), to obtain these parameters and their confidence intervals I use the Markov Chain Monte Carlo technique to obtain probability distributions for the parameter sets. For parameters where this distribution is symmetric or nearly symmetric extracting the parameter and a ~68% confidence interval is easy, just find the values at the 16th, 50th, and 84th percentiles and do a bit of subtraction.

However, for asymmetric distributions this is a bit trickier. Shown below is a corner plot from the one of the MCMC samplings. Marked on the single parameter probability distributions are the 16th, 50th, and 84th percentiles, while clearly for the (nearly) symmetric distributions, the best fit parameter and its errors are well determined, they aren't for the heavily skewed / asymmetric distributions.

Clearly the parameter I should pick as the "fit parameter" is the position of peak of the distribution.

My question is: How should I go about finding the confidence interval, especially when the peak is close to an edge of the distribution?

Note: This question is also posted on the Mathematics stack exchange, I just realized that this might be a better forum for such a question.

Answer 1

The description suggests you have MCMC samples. this leads me to expect that the model you fit was Bayesian. If that is the case then it does not make sense to look for confidence intervals for parameters. Instead, you could provide credible intervals; typically by stating the $\alpha/2$ and $(1-\alpha)/2$ empirical quantiles from your MCMC samples.

The asymmetry of the distributions does not change this. *Unless there is something quite special to say about whether your algorithm was likely to perform badly at the corners.

EDIT: the argmax of the posterior distribution is called the MAP estimate. This is what you are calling the 'fit value'. MAP estimates are great, but in Bayesian stats, it's not all about $MAP \pm 1.96 \sigma$. For instance you could easily take the mean or median of your samples rather than the mode. Basically your samples contain a lot more information than just the fit values. Consider how you would react if the hist below was of your posterior samples, would 3 be your 'fit value', or would 0? How would you summarise your samples?

Whatever prior you use, the posterior should be a probability distribution (caveat: search for 'improper priors').

Summary: you're on the right lines using quantiles of your samples (more care may be needed if you want to go beyond univariate summaries). Your issue is thinking that there's a problem with doing that; this likely caused by conceptual carry over from frequentist stats.