Suppose I have a binary outcome $Y$ and two binary covariates $X_1$ and $X_2$ with distribution $P(X_1, X_2)$ which I know. In addition to $P(X_1, X_2)$, I know $P(Y\mid X_1)$ and $P(Y\mid X_2)$. I would like to know how is $P(Y\mid X_1, X_2)$ bounded by the data I have at hand, and possibly know whether there is a natural visualisation of it.
I have checked previous questions here, here, and here. But unfortunately none of them provides an answer.
So far I have only been able to derive some trivial bounds on the probability. For example, if $P(y\mid x_1)\neq 0$ and $P(x_1, x_2)\neq 0$, then we can use $P(y\mid x_1) = \sum_{x_2} P(y\mid x_1, X_2=x_2)P(X_2=x_2\mid x_1)$ to conclude that $P(y\mid x_1, X_2=x_2)\neq 0$. But I haven't been able to come up with general formulas for the bounds on the full conditional.
The point of this question goes beyond three binary variables, I would even like to know whether there is a way to bound conditional probabilities based on arbitrary functions (say a conditional mean, instead of the conditional distribution) of marginalised distributions.
