# Modelling probabilities of a sum of binomials with different probabilities and trials

I have the following example data, where each row is an independent observation:

A B C Y
10 22 6 2
4 60 2 0
12 8 10 3
...

$$A$$, $$B$$, $$C$$ and $$Y$$ are all positive integers. The variables $$A$$, $$B$$ and $$C$$ represent the number of samples from a corresponding Bernoulli distribution: $$\textrm{Bernoulli}(p_A)$$, $$\textrm{Bernoulli}(p_B)$$, and $$\textrm{Bernoulli}(p_C)$$ respectively. Here, $$p_i \ne p_j, \forall i,j \in \{A, B, C\}, i \ne j$$ and $$p_i$$ is the same for each observation. The variable $$Y$$ represents the number of successes across all $$A + B + C$$ trials, that is:

$$Y = Y_A + Y_B + Y_C$$

where $$Y_A$$, $$Y_B$$ and $$Y_C$$ represent the number of successes for the $$A$$, $$B$$ and $$C$$ trials respectively (i.e. they are Binomially distributed). However, $$Y_A$$, $$Y_B$$ and $$Y_C$$ are unknown.

I am trying to model the $$p_i$$ parameter values in a Bayesian manner. The problem I see is that the $$Y_i$$ random variables have a different number of trials depending on the data observation. I take this to mean that rather than modelling $$Y_i \sim \textrm{Binomial}(n_i, p_i)$$ I would actually have to model $$Y_{ij} \sim \textrm{Binomial}(n_{ij}, p_i)$$, where $$j$$ corresponds to the index of the observation. Therefore, I would have a different random variable for each row in the data, which is not feasible given there would (in general) only be a single observation per distribution. Am I correct about this assumption? Regardless, how should I go about generating distributions for $$p_i$$ (perhaps by fitting a Beta distribution to each $$p_i$$)? Is it even possible, given the data?

I am very new to probabilistic modeling so I apologize if this problem is trivial or my understanding of key concepts is incorrect.

• What do you mean by "generating distributions for $p_i$"? Are you trying to do some kind of inference on the $p_i$? Maximum likelihood inference? Some kind of Bayesian inference, and if so, with what prior? Why is "not feasible" to have a different random variable for each row?
– D.W.
Commented Sep 19, 2022 at 4:07
• What does $p_i,p_j$ represent?
– D.W.
Commented Sep 19, 2022 at 4:31
• @D.W. Thanks for the help. I have edited the question to explain more clearly what I'm trying to do. Commented Sep 19, 2022 at 10:52
• Sums of independent binomials with different p's are called the poisson-binomial-distribution. You can peruse that tag! Commented Sep 19, 2022 at 14:10

Let $$A_i,B_i,C_i,Y_i$$ denote the values in the $$i$$th row. If $$p_A,p_B,p_C$$ are the same in all rows, then the appropriate model is
$$Y_i \sim \text{Bernoulli}(A_i,p_A) + \text{Bernoulli}(B_i,p_B) + \text{Bernoulli}(C_i,p_C).$$
If $$p_A,p_B,p_C$$ are not the same for all rows, then let $$p_{A,i},p_{B,i},p_{C,i}$$ denote the corresponding probabilities for all rows. Then the appropriate model is
$$Y_i \sim \text{Bernoulli}(A_i,p_{A,i}) + \text{Bernoulli}(B_i,p_{B,i}) + \text{Bernoulli}(C_i,p_{C,i}).$$
In the latter case, the data you have will not be enough to (uniquely) predict the $$p$$ values; there will be many possible $$p$$ values that are consistent with the observed data.