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.