# Bayesian a posteriori law of bernouilli law, from observed correlated bernouilli law

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.

• It's not really clear why you are putting a prior on $p$ given that each person will answer each of the two questions exactly once. So how do you expect to do better than something like $P(X_2|X_1)$, which can then be calculated in a Bayesian way from the rest of the students. – Alex R. May 3 '17 at 21:49
• Just to be sure, have you checked already the IRT models? en.m.wikipedia.org/wiki/Item_response_theory – mugen May 4 '17 at 1:25
• Alex: Thanks for the answer. I actually need more than 1 question, I chose 1 to simplify the problem. I shouldn't have. mugen: I read a little about IRT but it seems like it doesn't answer to my problem because I want various "Person locations" (IRT vocabulary) by persons. I want to modelize the impact of one ability (to answer question 1 for example) on the other (to answer question 2). Thanks anyway for the question and I will read more about IRT. – Rodolphe LAMPE May 5 '17 at 9:49