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Two recurrent difficult scenarios in probabilistic computations are dealing with multimodality and heavy-tailedness. Multimodality is difficult, but comes with some reasonable 'default' solutions, whereas heavy tails are perhaps less damaging, but require more care to resolve. In this blog post, I expand upon this distinction somewhat.
'Computation of Heavy-Tailed Measures'
@sp_monte_carlo, we often use/come up with specialized asymptomatic series expansions for computations involving heavy-tailed measures.
@sp_monte_carlo, here's an old, unfinished manuscript from nearly 10 years ago that demonstrates some of the computational work for marginalized Horseshoe (and other polynomial tail) priors in #sympy: https://tinyurl.com/hs-marginals. It uses a generalized G/H-function framework to cover entire classes of priors. Since there are broad theorems for producing asymptotic series of these special functions, it can work out pretty nicely.