beta-binomial as conjugate to hypergeometric

Question

According to the table of conjugate distributions on Wikipedia, the hypergeometric distribution has as conjugate prior a beta-binomial distribution, where the parameter of interest is "$M$, the number of target members." I interpret "target members" to mean, I am modeling as hypergeometric the number of blue balls in a sample from an urn containing $N$ total balls of which $M$ are blue, $N-M$ non-blue.

But then I cannot make sense of the claim of conjugacy. After the data is observed, say $b$ blue balls in the sample, then it is known that $M>b$. But a beta-binomial distribution is supported on $0,...,F$ (for some parameter $F$). So how can the posterior for $M$ also be beta-binomial?

Answer 1

The problem with the Wikipedia article and the reference therein (Fink D., 1997) is that there is some key information missing.

Specifically, the given posterior is for $M-x$ (i.e. the number of target individuals in the population shifted by the number observed in the sample), not for $M$. Furthermore, the posterior parameter corresponding to the number of observations is missing and should be $N-n$ (i.e. the population size minus the sample size). These two corrections fixes the support problem that you correctly noticed, as shown below.

Suppose that $0 \leq X \leq n$ is the number of target individuals in a sample of size $n$ from a population of size $N$ with $0 \leq M \leq N$ total target individuals.

Then, $X \sim \text{HG}(n, M, N)$ with support in $[\max(0, n-N+M), \min(n, M)]$.

If $M \sim \text{BB}(N, \alpha, \beta)$ is the prior distribution of $M$, the posterior distribution for $M - x$ is also Beta-Binomial-distributed: $$M - x\,|\,x,\alpha,\beta \sim \text{BB}(N-n, \alpha + x, \beta + n - x)$$

If you write the probability mass function for $M$ you will find @Tim's answer above.

As an illustration, for $N = 20$ and $n = 10$, let's assume a non-informative prior distribution for $M$ with $M \sim \text{BB}(N, .5, .5)$. Suppose that we observe $x = 9$.

library(extraDistr)
library(tidyverse)
N = 20
n = 10
a0 <- b0 <- .5
x <- 9
data.frame(
  m = 0:N
) %>% 
  mutate(
    prior = dbbinom(m, size = N, alpha = a0, beta = b0),
    post = dbbinom(m-x, size = N-n, a0+x, b0+n-x)
  ) %>% 
  gather(key, dens, -m) %>% 
  ggplot(aes(m, dens, col = key)) +
  geom_line() +
  geom_point()

^{Created on 2018-10-10 by the reprex package (v0.2.1)}

Note that the posterior support is correctly [x, N − n + x].

Dyer, D. and Pierce, R.L. (1993). On the Choice of the Prior Distribution in Hypergeometric Sampling. Communications in Statistics - Theory and Methods, 22(8), 2125-2146.

Answer 2

Hypergeometric distribution describes sampling without replacement from the urn containing $N$ balls, out of which $M \le N$ are target balls, let's say blue. The conjugate beta-binomial prior distribution leads to posterior distribution for unknown $M \in \{x,x+1,\dots,N-n+x\}$ in form

$$ f(M\mid x,N,\alpha,\beta) = {N-n \choose M-x} \frac{\Gamma(\alpha+M)\,\Gamma(\beta+N-x)\,\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+x)\,\Gamma(\beta+n-x)\,\Gamma(\alpha+\beta+N)} $$

as described in

Dyer, D. and Pierce, R.L. (1993). On the Choice of the Prior Distribution in Hypergeometric Sampling. Communications in Statistics - Theory and Methods, 22(8), 2125-2146.

Answer 3

As I cannot comment: the formula given by @Tom is not correct, the correct formula is :

$$ f(M\mid x,N,\alpha,\beta) = {N-n \choose M-x} \frac{\Gamma(\alpha+M)\,\Gamma(\beta+N-M)\,\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+x)\,\Gamma(\beta+n-x)\,\Gamma(\alpha+\beta+N)} $$

The formula is also wrong in the reference:

Dyer, D. and Pierce, R.L. (1993). On the Choice of the Prior Distribution in Hypergeometric Sampling. Communications in Statistics

• Theory and Methods, 22(8), 2125-2146.

at p. 2131. It is clearly a typo because on the same page, few lines below, that formula (in its correct form) is used to derive equation (3.1.3)