Beta Binomial Prediction

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.

Interactive Exploration

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

1. 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.
2. 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.
3. Set Future trials n to 3. The predictive horizon changes, and the possible counts range from 0 to 3.
4. 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.