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.