Beta-Binomial Distribution

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The beta-binomial distribution is an incredibly versatile tool. This simple Bayesian model has been used for decades to make accurate and informed predictions in the fields of epidemiology, intelligence testing and marketing.

Beta binomial distribution density function for several values of α and β. Image: Mschuma|Wikimedia Commons.

Its power comes from its shape parameters α > 0 and β > 0 which define the probability of success within a binomial framework – making it an especially powerful choice when faced with analytical dilemmas across many scientific disciplines.

  • For large values of α and β the distribution approaches a binomial distribution.
  • When α and β both equal 1, the distribution equals a discrete uniform distribution from 0 to n.
  • When n = 1, the distribution is the same as a Bernoulli distribution.

One 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.

The beta binomial distribution formula

Let’s say you have m items on an test, and each item is tested n times. The binomial distribution formula 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. But to create the formula, start with the PDF for the beta”

Formula for the beta distribution

And combine it with the binomial distribution formula to get a joint PDF:

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