Solved – Distribution of the sum of two independent Beta-Binomial variables

Consider two independent discrete random variables $y_1$ and $y_2$, both distributed with a Beta-Binomial distribution, with different number of successes $n_1$ and $n_2$ but the same parameters $a$ and $b$

$ p(y_1|n_1,a,b) = {n_1 choose y_1} dfrac{B(y_1 + a,n_1 -y_1 +b)}{B(a,b)} $

$ p(y_2|n_2,a,b) = {n_2 choose y_2} dfrac{B(y_2 + a,n_2 -y_2 +b)}{B(a,b)} $

Consider a discrete variable $Z = y_1 + y_2$. Is $Z$ distributed as well as a Beta-Binomial (with parameters $n_1+n_2$, $a'$ and $b'$?

I could not prove it in an analytic form so far, but I have been trying it out with some simulations, at least to check whether the assumption is wrong in some cases. Reparametrising $a$ and $b$ as $mu = dfrac{a}{a+b}$ and $rho = dfrac{1}{a+b+1}$, $y_1$ and $y_2$ have mean $mu n_1$ and $mu n_2$ respectively and variance $mu(1-mu)n_1(1 + (n_1-1)rho)$ and $mu(1-mu)n_2(1 + (n_2-1)rho)$ respectively.

Based on independence, the mean of $Z$ is $mu(n_1+n_2)$ and the variance of $Z$ is $mu(1-mu)(n_1(1 + (n_1-1)rho) + n_2(1 + (n_2-1)rho))$. If $Z$ was distributed according to a beta-binomial distribution, then it would have paramters $mu'$ and $rho'$, with $mu'= mu$ and

$rho' = dfrac{dfrac{n_1(1 + (n_1-1)rho) + n_2(1 + (n_2-1)rho)}{n_1+n_2} – 1}{n_1+n_2-1} = rho dfrac{n_1(n_1-1)+n_2(n_2-1)}{(n_1+n_2)(n_1+n_2 -1)} $

Here is some code to generate $Z$ as a sum of two independent Beta-Binomials (sorry about the code, R is not my main language)

n1 = 20 n2 = 50 mu = .6 k  = 20  p1  = rbeta(1e6,mu*k,(1-mu)*k) y1  = rbinom(1e6,n1,p1)  p2  = rbeta(1e6,mu*k,(1-mu)*k) y2  = rbinom(1e6,n2,p2)  z   = y1+y2  rho  = 1/(k+1) rho1 = rho*(n1*(n1-1)+n2*(n2-1))/((n1+n2)*(n1+n2-1)) k1   = 1/ rho1 - 1 p3  = rbeta(1e6,mu*k1,(1-mu)*k1) z1  = rbinom(1e6,n1+n2,p3)  print(c(var(z),var(z1))) plot(density(z,width= 3)) lines(density(z1,width = 3)) 

I have been trying this code for different values of $n_1$, $n_2$, $mu$ and $k$, but in all the cases the variances using the sum of two beta-binomials or an appropriately tuned beta-binomial are very similar (the densities look indistinguishable)

I know this question was posted a while ago, but I was looking for an answer to that question myself and stumbled upon this post. So here are my thoughts on it: in general, the answer is “no”. Consider an example of two iid random variable with beta-binomial distributions BB(1,1,n). BB(1,1,n) is the same as U(0,n) – the discrete uniform distribution on the interval [0,n]. The sum of two discrete uniforms is a triangular distribution (of discrete variety), which is not the same as the BB with parameters that you suggest.

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