Motivation for using a prior distribution with the same functional form as the likelihood

Question

When finding the functional form of the posterior distribution my textbook has suggested that when deciding on a prior distribution it should have the same functional form as the likelihood. Wondering why exactly this is the case ? Is it just for ease of analysis and interpretation or are there any other reasons ?

Answer 1

Mainly for convenience, but Bayesian computations can be challenging, so convenience may be a real issue.

If the prior and likelihood are 'conjugate' (mathematically compatible) then one can usually recognize the kernel of the posterior distribution. When that is possible, it is unnecessary to compute the integral in the denominator of Bayes' Theorem.

For example if the prior is $\mathsf{Beta}(1,1)$ and you observe $x$ successes in $n$ binomial trials, then the posterior distribution is easily recognized as $\mathsf{Beta}(x + 1, n - x + 1).$

In this situation observing 26 successes in 100 trials leads to a $\mathsf{Beta}(27, 75)$ posterior and a 95% posterior probability interval of $(0.184, 0,354)$ for the success probability. The only computation (shown here in R) is as follows:

qbeta(c(.025, .975), 27, 75)
## 0.1841349 0.3540134

If there were substantial prior information then the prior distribution might be $\mathsf{Beta}(5,8)$ and that would again lead to an obvious beta posterior distribution. One could have chosen a normal distribution with mean and variance matching those of $\mathsf{Beta}(5,8)$ (and preferably truncated to have support $(0,1)$). The resulting posterior probability interval wouldn't be much different, but you'd have an ugly (probably numerical) integration to do before you could compute the probability interval.

Often the choice of a prior is not precisely indicated. If feasible, one might as well choose a prior distribution that matches prior information closely and makes life easy.