# Censored multinomial with different, observed censorship pools

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!