Consider a random variable $x$ and $\theta$. Suppose $f(\theta)$ and $f(\theta|x)$ are given. Suppose, in addition, that $f(\theta|x)$ satisfies the strict monotone likelihood ratio property. That is, for every $\theta >\theta'$ and $x > x'$, we have $f(\theta'|x')f(\theta|x)>f(\theta|x')f(\theta'|x).$ Will they uniquely determine the joint distribution $f(x, \theta)$?
Determination of the joint distribution
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probability