# 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?

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Is this an homework?! The question sounds neither correct nor complete: the "evidence" is an additional item of information, not an observation, since the event $B>A$ occurs with probability $p$. Furthermore, this is not a sufficient item to determine the joint distribution of $(A,B)$. – Xi'an Jan 19 '12 at 17:53
There are two types of artefact for which, from archaeological evidence, we would estimate a date expressed by two (given) probability distributions. Subsequently it is discovered that the two types are found juxtaposed at a dig in such a way that A is 95% likely to be of earlier provenance than B. How should our probability distributions change to reflect this? – Pete Jan 19 '12 at 23:52
Interesting context, but this does not change the mathematical issue. Your additional piece of information contributes to the definition of a joint distribution on (A,B), but (again) this is not enough for defining by itself a joint. This is not a Bayesian updating issue as far as I can judge. If you do need a joint, you can pick it within a copula and imposing the 95% $A>B$ constraint. – Xi'an Jan 20 '12 at 5:43
This is alas getting more confusing: are $f_A$ and $f_B$ priors on the expectations of $A$ and $B$, respectively? Even if this is the case, you are not in a position to run a Bayesian estimation if your sole evidence is that "b>a with probability $p$" since, again this is not an event taking place in the observation space. – Xi'an Jan 21 '12 at 17:03
Yes, the $f_A$ and $f_B$ are representations of the expected chronology of the two species individually - before their mutual relationship is discovered or taken into account. I'm not completely sure I understand the significance of the confidence that b>a not being an event in the observation space, but moving away from the statistical model, it seems clear that the chronological expectations must change as a result of the discovery. Are you saying that it is not possible to quantify the extent to which they change? – Pete Jan 22 '12 at 10:38
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,
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 $A_i>B_i$.