# Why the mixtures of conjugate priors is important?

I have a questions about the mixture of conjugate priors. I learnt and say the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such important, how are we going to apply it when we are doing Bayesian analysis.

To be more specific, one theorem from Diaconis and Ylivisaker 1985 illustrated a theorem as this:

Given a sampling model $p(y|\theta)$ from an exponential family, any prior distribution can be approximated by a finite mixture of conjugate prior distributions.

More specifically, given prior $p(\theta)=\int p(\theta|\omega)p(\omega)d\omega$, we can derive the posterior:

$p(\theta|Y)\propto\int p(Y|\theta)p(\theta|\omega)p(\omega)d\omega\propto\int \frac{p(Y|\theta)p(\theta|\omega)}{p(Y|\omega)}p(Y|\omega)p(\omega)d\omega\propto \int p(\theta|Y, \omega)p(Y|\omega)p(\omega)d\omega$

Therefore,

$p(\theta|Y)=\frac{\int p(\theta|Y, \omega)p(Y|\omega)p(\omega)d\omega}{\int p(Y|\omega)p(\omega)d\omega}$

• This is not an answer to your question, but it is good to remember that in many cases you do not have to use conjugate priors for sampling (check here). – Tim Dec 12 '14 at 8:47
• The theorem you quote is not true. The version you describe is about hierarchical priors, not conjugate priors. Please rephrase your question correctly. – Xi'an Dec 12 '14 at 10:35
• @Xi'an Thanks. This quote originates from the paper <statistics.stanford.edu/sites/default/files/EFS%20NSF%20207.pdf>. It is on the bottom of page 13. – Shijia Bian Dec 12 '14 at 14:15
• Oh, you forgot the "approximation" and the "finite" in the statement!!! "Any prior can be approximated by a finite mixture of conjugate priors" is the right quote, with the approximation not working in terms of tail behaviour. – Xi'an Dec 12 '14 at 16:05
• @Xi'an may I also have another question? Why should we always emphasize on "finite" mixture model? In other words, is there infinite mixture model? – Shijia Bian Dec 12 '14 at 20:02
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