PPS (EDITED to rule out Interpolated): the copula example (and the original post) does point to a rather general solution, provided one can find a pymc distribution that approximate the marginals well enough and has the icdf method implemented. Unfortunately, this rules out Interpolated since it does not implements icdf. So here is how it would go:
- After running modelA, identify marginal distributions that describe the posterior of modelA well – in the general case, that could be the
Interpolateddistribution, but it lacks theicdfmethod so you need to find another distribution such asLogNormal. - transform the variables using the
cdfmethod of that distriubtion, to obtain uniformly distributed variables - transform again with the normal distribution’s inverse cdf method (ppf in scipy, icdf in pymc)
- fit a multivariate normal distribution to the result of 3 => the resulting
muandcov(orchol) will be your prior in modelB
Note the steps 1-4 need not be done in pymc.
And in modelB (needs to be done within a pymc model):
5. Define a MvNormal distribution with the mu and chol parameters obtained in 4
6. Transform the variables back to their original, custom shape by reverting 3. (use the normal distribution cdf method) and 4. (use the Interpolated distribution icdf method)
It feels a bit involved, and I wonder if 5) and 6) could cause convergence issues (assuming they are tracked in the pymc graph and NUTS has to calculated gradients on these) – the linked notebook above mentions convergence issues --, but it could be tried. In practical settings, it is probably best to apply the transformations to only these variables that have a long tail and are clearly not normally distributed, and leave the rest untransformed.
EDIT PPPS: in the case of a LogNormal distribution, it’s enough to just take the logarithm of that variable and use that in the sampling, without the intermediate steps of converting to a uniform distribution.