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Is there a prior that's commonly used for "index" or "score" type variables that are user-defined as a weighted sum of a small number of variables (sometimes with pre-defined interaction contributions)?

I know the gamma, inverse gamma, binomial, negative binomial, poisson, etc distributions are strictly positive, but they correspond to processes that don't describe something like an index/score type variables well. Distributions of index/score type of values are usually overdispersed relative to these distributions.

I could construct my own mixture distribution, but I was wondering if there is are standard practices for this kind of data.

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