Beta Binomial Prediction

Compare a beta-mixture predictive count with a binomial count having the same mean probability.

Beta Binomial Prediction

Mixture Models and Overdispersion

When modeling a series of Bernoulli trials, a standard binomial distribution assumes the success probability is exactly known and identical across all trials. In many real-world predictive scenarios, the underlying probability itself is uncertain and governed by a prior distribution, often modeled with a Beta distribution. The Beta-Binomial distribution emerges when we marginalize out this uncertain probability. The shared uncertainty in the parameter introduces positive correlation among the trials, leading to "overdispersion" — greater variance than a binomial model with the equivalent mean probability would predict.

Analytic Setup and an Independently Worked Example

Let a shared success probability P follow a continuous Beta(a, b) distribution. Conditionally on P = p, let the count K of successes in n trials follow a Binomial(n, p) distribution. Integrating out p gives the marginal probability mass function for K = k:

P(K = k) = C(n, k) * B(a + k, b + n - k) / B(a, b)

where B is the Beta function. The mean of this predictive distribution is n * a / (a + b). Its variance is n * a * b * (a + b + n) / ((a + b)² * (a + b + 1)), which for n greater than 1 exceeds the variance of a simple Binomial plug-in distribution evaluated at the expected probability p = a / (a + b).

Consider an original worked example where a = 2, b = 3, and n = 3.

The expected success probability is a / (a + b) = 2 / 5 = 0.4. The expected number of successes is n * (2/5) = 3 * 0.4 = 1.2.
The Beta-Binomial variance is (3 * 2 * 3 * (2 + 3 + 3)) / ((2 + 3)² * (2 + 3 + 1)) = (18 * 8) / (25 * 6) = 144 / 150 = 0.96.
For comparison, a standard Binomial with n = 3 and p = 0.4 would have variance 3 * 0.4 * 0.6 = 0.72. The Beta-Binomial is distinctly overdispersed due to the uncertainty in P.
We can calculate the specific probability of observing exactly k = 2 successes. We need B(2+2, 3+3-2) = B(4, 4) and B(2, 3). For integers, B(x, y) = (x-1)! (y-1)! / (x+y-1)!. So B(4, 4) = (3! 3!) / 7! = (6 * 6) / 5040 = 1 / 140. Also B(2, 3) = (1! 2!) / 4! = 2 / 24 = 1 / 12.
The PMF is C(3, 2) * (1 / 140) / (1 / 12) = 3 * (12 / 140) = 36 / 140 = 9 / 35 ≈ 0.2571.

Interactive Exploration

You can test these properties interactively in the Beta Binomial Prediction module.

Set Future trials n to 4, Shape a to 1 and Shape b to 1. This represents a uniform prior on the success probability. Observe the uniform Beta-Binomial PMF (probability 0.2 across all counts) compared to the peaked Same-Mean Binomial. Note the Beta-Binomial variance is 2.0 while the Binomial variance is 1.0.
Change Shape a to 2 and Shape b to 2. This creates a prior centered at 0.5. The Beta-Binomial variance drops to 1.6, but is still greater than the Binomial's 1.0.
Set Future trials n to 3. The predictive horizon changes, and the possible counts range from 0 to 3.
Set Success count k to 3 to probe this specific outcome. The exact probability (PMF at k=3) will update. If you increase k beyond n (e.g., k=4), the PMF correctly evaluates to 0.

Model Limitations

The Beta-Binomial model assumes that a single unobserved probability P governs all n trials simultaneously, rather than drawing a fresh independent p for each trial. If the probability were re-drawn independently for each trial, the marginal count distribution would simply revert to a Binomial distribution with p = a / (a + b).

Additionally, this analytical model assumes the prior follows specifically a Beta distribution, making it a conjugate prior. Other bounded distributions on [0,1] could be used for the uncertainty in P, but they would not result in this exact, closed-form predictive PMF. The positive shape bounds are small integers (1 to 4). On the declared trial grid, factorial products stay within exact safe-integer range, while their ratios and displayed probabilities use ordinary floating-point arithmetic. At n = 0 or n = 1 the predictive and same-mean binomial variances are equal.

References and Further Reading

For more information on the Beta-Binomial distribution, you can refer to the following official documentation:

SciPy Community (2026). scipy.stats.betabinom. Official SciPy reference documentation on the Beta-Binomial distribution and its probability mass function derived from the Beta function.

Beta Binomial Prediction