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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

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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.

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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.

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