Equivalent of Conjugate Priors for Marginal Probability Distributions

Question

In probability, there are nice “conjugate prior” distributions that enable closed-form Bayesian updating – e.g. if you have a Normal likelihood and Normal prior (on the mean parameter), you get a Normal posterior.

Is there an equivalent concept for “nice” (e.g. closed form) marginal distributions? For example, if I have a “A”-distributed likelihood for random variable X (conditional on parameter M) and a “B”-distributed prior on parameter M, I can integrate out the uncertainty of M to get the unconditional distribution of X -- let’s say that it is “C”-distributed. My question is: are there well-known pairings of “A” and “B” that lead to well-known “C” here?

• The only example I can think of is for a Normal likelihood and Normal prior on the mean, but in particular I was wondering if there were example of this for “heavy”-tailed distributions like Student-t, Pareto, etc.
• Obviously, you can just flip Bayes Rule to calculate the Marginal Distribution given the likelihood, the prior, and the posterior --- but that doesn’t always result in a well known probability distribution.

Answer 1

The ones I encounter the most in practice are:

• Poisson rates with between subject variation on rates following a Gamma distribution: Negative binomial when you integrate out the random subject effect (that version of the negative binomial distribution is used a lot for negative-binomial regression of overdispersed count data)
• Binomial with between subject variation on probabilities following a Beta distribution: Betabinomial when you integrate out the random subject effect
• Normal distribution with inverse-gamma distribution on the variance: Student-t distribution when you integrate out the uncertainty on the variance

You can find some more examples in the "posterior predictive distribution" column of the Wikipedia table of conjugate prior distributions - unsurprisingly the topics are closely linked.