Bayesian inference for a conditional probability

Question

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

Trial 1: (head,head), Trial 2: (head,no tossing), Trial 3: (tail, tail), Trial 4: (tail,head) and so on.

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.

Answer 1

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.