The Beta-Binomial Distribution

As in the previous section, let $X$ have the beta $(r, s)$ prior, and given $X = p$ let the $S_n$ be the number of heads in the first $n$ tosses of a $p$-coin.

All the calculations we carried out in the previous section were under the condition that $S_n = k$, but we never needed to find the probability of this event. It was part of the constant that made the posterior density of $X$ integrate to 1.

We can now find $P(S_n = k)$ by writing the posterior density in two ways:

  • By recalling that it is the beta $(r+k, s+n-k)$ density:
$$ f_{X \vert S_n=k} (p) ~ = ~ C(r+k, s+n-k)p^{r+k-1}(1-p)^{s+n-k-1}, ~~~~ 0 < p < 1 $$
  • By using Bayes' Rule:
$$ f_{X \vert S_n=k} (p) ~ = ~ \frac{C(r, s) p^{r-1}(1-p)^{s-1} \cdot \binom{n}{k} p^k (1-p)^{n-k}}{P(S_n = k)}, ~~~~ 0 < p < 1 $$

Now equate constants:

$$ \frac{C(r, s) \binom{n}{k}}{P(S_n = k)} ~ = ~ C(r+k, s+n-k) $$

Beta-Binomial Probabilities

So for $k$ in the range 0 through $n$,

$$ P(S_n = k) ~ = ~ \binom{n}{k} \frac{C(r, s)}{C(r+k, s+n-k)} $$

where $C(r,s)$ is the constant in the beta $(r, s)$ density, given by

$$ C(r, s) ~ = ~ \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} $$

This discrete distribution is called the beta-binomial distribution with parameters $r$, $s$, and $n$. It is the distribution of the number of heads in $n$ tosses of a coin that lands heads with a probability picked according to the beta $(r, s)$ distribution.

One $(r, s)$ pair is particularly interesting: $r = s = 1$. That's the case when $X$ has the uniform prior. The distribution of $S_n$ reduces to

$$ P(S_n = k ) ~ = ~ \frac{n!}{k!(n-k)!} \cdot \frac{1!}{0!0!} \cdot \frac{k!(n-k)!}{(n+1)!} ~ = ~ \frac{1}{n+1} $$

There's no $k$ in the answer! The conclusion is that if you choose $p$ uniformly between 0 and 1 and toss a $p$-coin $n$ times, the distribution of the number of heads is uniform on $\{ 0, 1, 2, \ldots, n\}$.

If you choose $p$ uniformly between 0 and 1, then for the conditional distribution of $S_n$ given that $p$ was the selected value is binomial $(n, p)$. But the unconditional distribution of $S_n$ is uniform.

Checking by Integration

If you prefer, you can find the distribution of $S_n$ directly, by conditioning on $X$. $$ \begin{align*} P(S_n = k) ~ &= \int_0^1 P(S_n = k \mid X = p)f_X(p)dp \\ \\ &= ~ \int_0^1 \binom{n}{k} p^k(1-p)^{n-k}C(r, s)p^{r-1}(1-p)^{s-1}dp \\ \\ &= ~ \binom{n}{k} C(r, s) \int_0^1 p^{r+k-1}(1-p)^{s+n-k-1} dp \\ \\ &= ~ \binom{n}{k} C(r, s) \frac{1}{C(r+k, s+n-k)} \end{align*} $$

Expectation

Given $X = p$, the conditional distribution of $S_n$ is binomial $(n, p)$. Therefore

$$ E(S_n \mid X = p) ~ = ~ np $$

or, equivalently,

$$ E(S_n \mid X) ~ = ~ nX $$

By iteration,

$$ E(S_n) ~ = ~ E(nX) ~ = ~ nE(X) ~ = ~ n\frac{r}{r+s} $$

The expected proportion of heads in $n$ tosses is

$$ E\big{(} \frac{S_n}{n} \big{)} ~ = ~ \frac{r}{r+s} $$

which is the expectation of the prior distribution of $X$.

In the next section we will examine the long run behavior of this random proportion.

Endnote

The unconditional probability $P(S_n = k)$ appeared in the denominator of our calculation of the posterior density of $X$ given $S_n$. Because of the simplifications that result from using conjugate priors, we were able to calculate the denominator in a couple of different ways. But often the calculation can be intractable, especially in high dimensional settings. Methods of dealing with this problem are covered in more advanced courses.