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In the problem I'm working on, I'm trying to infer the proportions of three types of object $A, B, C$ in a population. I'll use $p_A$, $p_B$, and $p_C$ to refer to the proportions. The data I get to see is censored so that sometimes I only observe that an observation is $A$ or not $A$, and sometimes I only observe that it's $B$ or not $B$. However, I always know which of the two censorships is happening. So basically I see two random variables:

$$ x_A \sim \mathrm{Bin}(p_A, N_1) \\ x_B \sim \mathrm{Bin}(p_B, N_2) $$

(The $N_1$ and $N_2$ variables are uninformative.)

I'd like to get ML estimates of $p_\cdot$, but I'm having trouble with the optimization. My intuition is that this must be identifiable, because I have two likelihood equations to estimate parameters with two degrees of freedom.

Using Lagrange multipliers, I've convinced myself that the solution has to satisfy the following equations:

$$ \frac{x_A}{p_A} + \frac{N_2 - x_B}{p_A + p_C} = \frac{N_1 - x_A}{p_B + p_C} + \frac{x_B}{p_B} = \frac{N_1 - x_A}{p_B + p_C} + \frac{N_2 - x_B}{p_A + p_C} $$

However, I'm not sure if that's even particularly helpful. The log-likelihood is also convex, so local optimization methods could also work. I don't know how to enforce the constraint that $p_A + p_B + p_C = 1$ in a local optimization method though.

I'd appreciate any help toward a solution, either in terms of optimization algorithms or an explicit formula. Thanks!

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