Bayesian updating of continuous variables given mutual information

Question

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.

Answer 1

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,

1. Unless you specify the family of joint distributions from the start, you do not have enough information in your problem to find the joint.
2. 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.
3. 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 $A_i>B_i$.