# Estimating parameter from conditionals of non-independent data

#### VelocideX

##### New Member
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.