# In Bayesian statistics, must I use a marginalized prior in conjunction with a marginalized distribution?// [closed]

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

## closed as off-topic by Ricardo Andrade, Joonas Ilmavirta, Alex Degtyarev, Stefan Kohl, LuciaMay 14 '15 at 13:23

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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.

Your full posterior distribution will be proportional to g(x,y,z) k(x,y,z). To find the posterior marginal distribution on x, you have to integrate the product g(x,y,z) k(x,y,z) with respect to y and z. You can't integrate g with respect to y and z first before introducing k.