# Bayesian inference for a conditional probability

I'm simplifying my research question and want to know whether the question can be properly modeled or not.

Suppose we have two coins $$X_1,X_2$$ and assume that the outcomes are possibly correlated. Denote $$p_i=\mathbb P[X_i=1]$$ and $$p_{12}=\mathbb P[X_1=1\wedge X_2=1]$$.

We have to estimate $$p_i$$, $$p_{ij}$$, $$p_{X_1|X_2}$$ and $$p_{X_2|X_1}$$ from data, a table of heads/tails of each trial. What is more is that, in some trials, only one coin is tossed. For example, the data we have is

Would there be a prior distiribution(such as the Beta conjugate in independent variable casees) that we can handle Bayesian inference for this bivariate Bernoulli trials?

Any comments will be very much appreciated.

Since you've got a known number of outcomes (9, if I'm counting correctly, which arguably is not very many), you could model this data in a general way as a categorical distribution. In that case, you would have a vector of probabilities, each corresponding to the discrete outcomes {(head, head), (head, tail), (head, none), (tail, head), (tail, tail), (tail, none), (none, head), (none, tail), (none, none)}. In that case, you can use a Dirichlet as your conjugate prior distribution.