# Why is the mixtures of conjugate priors important?

I have questions about the mixture of conjugate priors. I learned and saw 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, 2014 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. Dec 12, 2014 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. Dec 12, 2014 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. Dec 12, 2014 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? Dec 12, 2014 at 20:02