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Consider a variable $x$ with prior distribution $f(x)$ over the support $X$. Suppose we observe a signal $s$. Then, by using the likelihood $L(s|x)$ of observing this signal for a given $x$, we can update our prior as follows.

$f'(x) = \dfrac{L(s|x)f(x)}{\int\limits_{x\in X} L(s|x')f(x')dx'}$

However, I am having trouble doing the same when the signal depends on two independent variables $x$ and $y$ with prior distributions $f(x)$ and $g(y)$. We now have a similar likelihood of observing a signal $s$ for a given $x$ and $y$ given by $L(s|x,y)$. Now, how do we simultaneously update $f(x)$ and $g(y)$?

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Let $h(x,y)$ denote the joint distribution of $x$ and $y$. Then

$$ f'(x) = \frac{\int_{y \in Y} L(s|x,y) h(x,y) dy}{\int_{x \in X}\int_{y \in Y} L(s|x,y) h(x,y) dy dx}$$