In any Metropolis-Hastings approach, you need some kind of likelihood function that you can use to decide if you stay where you are in the chain or move to the new proposed value. I've been using simply computing the likelihood of getting the observed block average given the proposed mean and observed variance.
Here's the kicker, though: these are heavy tailed distributions. The variance may not to exist. Of course it will always exist for a finite sample, but if the underlying data truly is heavy tailed, that's a rather bogus measure.
So, now I'm thinking about how you compute a likelihood using some other measure of spread that can be reasonably estimated. Interquartile range is one obvious one, though it suffers from the fact that it's not particularly helpful in assessing the mean, particularly if the distribution is multi-modal. There might be some real math to be done here.
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