Estimating parameter from conditionals of non-independent data

#1
The context of this problem is the estimation of the distribution of a parameter \(v\) given sets of data \(A\) and \(B\), where \(A\) and \(B\) are not independent.

Suppose I know \(P(v | A)\) and \(P(v | B)\). Both of these distributions are definitely not normal (the answer should not make any assumptions about the functional form here)

Suppose I also know that \(A\) and \(B\) are not independent. In particular, the data come in pairs \((a_i, b_i)\). The exact relationship between \(a_i\) and \(b_i\) is not known but one can suppose that they are linearly correlated i.e. \(a_i \sim N(f + gb_i, \sigma_a^2)\), or alternatively \(b_i \sim N(j + ka_i, \sigma_b^2)\) . However, it should not be assumed that the data sets \(A\) and \(B\) are normally distributed (either jointly or independently).


The question: how do I work out \(P(v | A, B)\)?

The obvious approach is to try : \(P(v | A, B) \propto P(A | v, B) P (v | B)\), however I can't see how to calculate the first term with only the knowledge above.

Another approach is to try \(P(v | A, B) \propto P(A, B | v) P(v) = P(A | B, v) P(B | v)\) which doesn't help either.

The inter-relationship of \(A\) and \(B\) suggests that \(P(A | B, f, g) = \prod_i P(a_i | b_i, f, g, \sigma)\). The \(f\), \(g\) and \(\sigma\) can be integrated out if needed (with appropriate priors) to get \(P(A | B)\). But I can't see how to introduce the relationship between \(A\) and \(\nu\) into this to get to \(P(A | \nu, B)\).

One approach I have considered is
\(P(\nu | A, B) = \int P(\nu, X | A, B) \delta(X - \nu)\,dX \stackrel{?}{=} \int P(\nu | A)P(X | B) \delta(X - \nu)\,dX\)
\( = P(\nu|A)P(\nu|B)\) but I'm almost certain that the second step isn't valid.

Any help greatly appreciated.