# Beta-Binomial Distribution

< Probability distributions list < Beta-binomial distribution

The beta-binomial distribution is a discrete probability distribution that uses the beta distribution as a prior distribution for the probability of success in a binomial experiment. While the binomial distribution has fixed probabilities, the beta-binomial probabilities can vary from trial to trial, which makes it a more flexible distribution than the binomial.

This simple Bayesian model has been used for decades to make informed predictions in fields such as cognitive science, epidemiology, intelligence testing and marketing.

## Beta-binomial distribution

The probability mass function (PMF) for the beta-binomial distribution is:

Where x ∈ { 0, …, n }.

Two shape parameters α > 0 and β > 0 define the probability of success.

• For large values of α and β the distribution approaches a binomial distribution. In other words, the binomial distribution is the limiting distribution.
• When α and β are both equal to 1, the distribution is a discrete uniform distribution from 0 to n. This is because when the beta function has α = 1 and β = 1, it simply equals 1, meaning that probability of getting any number of successes is equal.
• When n = 1, the distribution is equal to a Bernoulli distribution (which models a single trial) with p chosen from a beta distribution, which has mean α/(α + β).

where 3F2(a;b;x) is the generalized hypergeometric function 3F2(1, –x, n x + β; n x + 1, 1 – x α; 1).

The mean of the beta-binomial distribution is / (α + β).

The variance is the product of two terms :

• The first term, nαβ / (α + β)² , is the variance for a binomial distribution with the same expected value as the beta-binomial distribution.
• The second term, (α + β + n) / (α + β + 1), is a multiplier greater than 1 for n > 1.

This means that a beta-binomial distribution with n > 1 always has a larger variance than a binomial distribution with the same expected value and number of trials.

## Deriving the beta-binomial distribution formula

The beta-binomial(n, α, β) distribution is generated by choosing probability p for a binomial(n, p) distribution from a beta(α, β) distribution. You can think of it as a combination of both the binomial and beta distributions.

Let’s say you have m items on an test, and each item is tested n times. The binomial distribution PMF is:

• P = binomial probability,
• xi = total number of “successes” (pass or fail, heads or tails etc.) for the ith trial,
• pi = probability of a success on an individual trial,
• n = number of trials.

You can also think of p as being randomly drawn from a beta distribution. To create the beta-binomial formula, combine the binomial distribution PMF with the PMF for the beta distribution

to get a joint PMF:

Which can also be written (using Beta distribution properties) as:

## Difference between the binomial and beta-binomial distribution

The major difference between a binomial distribution and beta-binomial distribution is that in a binomial distribution, p is fixed for a set number of trials; in a beta-binomial, p is not fixed and changes from trial to trial. One benefit is that the beta-binomial distribution can be used to model data that is overdispersed, which means that the variance is greater than the mean. This can happen in a many situations, such as when there are more extreme values in a dataset than would be expected if the data followed a normal distribution.

## References

 Nschuma, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

 Beta-binomial (n, α, β) distribution. Retrieved July 14, 2023 from: https://www.acsu.buffalo.edu/~adamcunn/probability/betabinomial.html

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