Is there any way to construct an expected conditional probability distribution of the form p(x|(y,z)) if I am starting with p(x|y) and p(x|z)? All variables are categorical.
My specific problem deals with a DNA multiple-sequence alignment. Abstractly, I have a two-dimensional dataset, where each position can hold one of four values {G,A,C,T}. The rows are genomes and the columns are aligned positions in the different genomes. I want a way to judge if the observed value is surprising given what I know about the row and column in which it is observed. My intention is to identify regions of a chromosome (a portion of a row, ~1000 columns wide) where the observed values would not have been predicted based on the characteristics of the row and the specific columns. I can easily infer the probability of observing a given outcome (e.g. x = G) for both the row and the column, but I don't know how I would merge those expectations into a single set of expected frequencies.
I would like to say that I am assuming that Y and Z are independent, but I don't know if that's meaningful in this scenario. One obvious constraint is that if a particular outcome is forbidden by one of the predictors, then it cannot occur at the intersection of the two predictors. For example, if the column is uniform (e.g. the genomes are monomorphic) then it doesn't matter what the probability distribution is for the row in question.
I feel like I'm missing something obvious, and I have an intuitive solution but am unable to prove it to myself. Any suggestions would be appreciated.