Bayesian updating of continuous variables given mutual information I have very little stats training, so this may be a very obvious and boring question, in which case I apologise.
Given two real-valued continuous random variables A and B, and given prior probability distributions $f_A$ and $f_B$ for each, how do I perform a Bayesian update when given the extra evidence that b>a with probability p?
Thanks in advance.
 A: Given two random variables, $A$ and $B$, with marginal densities $f_A$ and $f_B$, there are an infinite number of joint distributions compatible with those marginals densities and with the fact that $A>B$ with probability $p$. For instance, if $F_A$ and $F_B$ denote the cdfs associated with $f_A$ and $f_B$, i.e.
$$
F_A(a) = \int_{-\infty}^a f_A(x)\text{d}x
\quad
F_B(b) = \int_{-\infty}^b f_B(x)\text{d}x\,,
$$
then the joint distribution with cdf
$$
P(A<a,B<b) = \dfrac{F_A(a)F_B(b)}{1+\varrho (1-F_A(a))(1-F_B(b))}
$$
is compatible with the marginal densities and, for a proper choice of $\varrho$ leads to $P(A>B)=p$ (unless the supports of $f_A$ and $f_B$ prevent the event to occur). Thus,


*

*Unless you specify the family of joint distributions from the start, you do not have enough information in your problem to find the joint.

*The fact that $P(A>B)=p$ is more an item of information about the joint distribution of $(A,B)$, than about the parameters of $f_A$ and $f_B$. In any case, there is no Bayesian component in the problem. 

*An example of a Bayesian problem would be an inference about $\varrho$ when observing $n$ pairs $(A_i,B_i)$. Or even observing that out of $n$ pairs $(A_i,B_i)$, $m$ are such that $A_i>B_i$.

