Suppose that you have $N$ people passing an exam having 2 yes/no questions. After seeing the results of $N-1$ people and the result of the first question of the $Nth$ person, I want to know what I can infer for the second question.
I modelize the question’s results by two bernoulli laws $X_1 = B(p_1)$ and $X_2 = B(p_2)$. Each person has his own $p_1, p_2$ but we think that they’re linked. I’m looking for a way to use this dependence …
So far, I have chosen an a priori law on $p_1, p_2$ : the $beta(1,1)$ function for both. We know that if a person answers with $S$ success and $E$ errors for the first question, the a posteriori law will be $beta(1+S, 1+E)$. But we don’t know anything about how it changes $p_2$. My guess is that $p_2$ could be something like $beta(1+kS, 1+kE)$ with $k$ a coefficient representing how much the first question and second question are correlated. This intuition comes from the facts that if the questions are independent then the a posteriori law of $p_2$ is still $beta(1,1)$ and if they are perfectly correlated, the a posteriori law of $p_2$ is $beta(1+S, 1+E)$.
What about in between ? Any references/books/etc would be greatly appreciated.