broverman credibility ch 5 pr 10

[shakeyhands]

The question involves model dist = poisson and prior = mixture of exponentials. He states in the solution that the posterior will be a mixture of exponentials as well and skips the ugly calculus. This isnt obvious to me: Is this a special property of gamma distribtution mixtures? Is this true in the general case, for example is the posterior dist of model binomial, prior mixed beta, going to be mixed beta and likewise for all conjugate prior mixtures? Am i missing something simple?

[sam_broverman]

Shakeyhands,

With a prior density pi(lambda) and a model density f(x | lambda)
we get the joint density of x and lambda as
f(x,lambda) = f(x|lambda) pi(lambda) .

We also know that the posterior density (as a function of lambda) is proportional to the joint density, and this is what allows us
to know that a gamma prior and Poisson model combine to
result in a gamma posterior ( because f(x,lambda) is proportional
to a gamma dist with variable lambda).

Suppose that the prior is a mixture of two (or an number of ) gammas,
say a1 pi1(lambda) + a2 pi2(lambda) (where a1 and a2 are the
mixing weights). Then with Poisson model dist, the posterior will
be proportional to

f(x,lambda) = f(x|lambda) [ a1 pi1 (lambda) + a2 pi2(lambda) ]
= a1 f(x|lambda) pi1 (lambda) + a2 f(x|lambda) pi2 (lambda) .

Each of f(x|lambda) pi1 (lambda) and f(x|lambda) pi2 (lambda)
is proportional to a gamma dist, so the posterior is prop to
a linear combination of gammas, which makes it a mixture of
gamma distributions.

This idea works with any prior-model combinations that results
in an identifiable posterior. Your example of beta prior and
binomial model would work the same way.

Having said all this, it will take some algebra to actually
identify the mixing weights in the posterior distribution.

Sam Broverman
2brove@rogers.com

[shakeyhands]

thanks for the help!