Suppose I have some sampling distribution g(x,y,z) which has been marginalized over some variables (say y and z) giving us the marginal distribution which we'll call gx(x).


Suppose I now wish to use Bayes Theorem but on the marginalized distribution to obtain the posterior marginal distribution. Suppose I also know the a good prior for all the variables, call it k(x,y,z).... To use Bayes theorem on the marginalized distribution, must I also marginalize the prior? Or does it makes sense to use the full prior, which of course makes, the answer depend on 



In Bayesian statistics, must I use a marginalized prior in conjunction with a marginalized distribution?//
No, you cannot marginalize the prior and then multiply the marginal with g(x). Here is why:


The correct way to use Bayes theorem is to do the following (also suggested by John):


$g\_{p1}(x,y,z) \propto g(x,y,z) k(x,y,z)$


Thus,


$g\_{p1}(x) \propto \int\_{y,z} \bigl(g(x,y,z) k(x,y,z) \bigr)$


Your want to do the following:


$g\_{p2}(x) \propto \int\_{y,z} \bigl(g(x,y,z) \bigr) \int\_{y,z} \bigl(k(x,y,z) \bigr)$


In general, $g\_{p1}(x)$ and $g\_{p2}(x)$ will not be identical. 


