Beta–Geometric Waiting Time

The Beta–Geometric mixture models the waiting time until the first success when a single unknown success probability P is drawn once from a Beta(a, b) prior and held fixed across all trials of that experiment.

Concept & Intuition

If the success probability p were known and fixed, the trial index T of the first success (starting at t = 1) would follow a standard Geometric distribution with constant hazard P(T = t | T ≥ t) = p. When P is drawn once from Beta(a, b) at the start of the waiting experiment, each observed failure updates the conditional distribution of P downward to Beta(a, b + t - 1).

Consequently, the hazard rate decreases monotonically with t, producing a heavier waiting-time tail than a plug-in geometric distribution using the mean success probability E[P] = a / (a + b).

Mathematical Formulation

Let T be the trial index of the first success (t = 1, 2, ...). Averaging the conditional geometric mass P(T = t | P = p) = p(1 - p)^t-1 over P ~ Beta(a, b) yields:

PMF: P(T = t) = B(a + 1, b + t - 1) / B(a, b)

Survival before trial t: P(T ≥ t) = B(a, b + t - 1) / B(a, b)

Hazard at trial t: h(t) = P(T = t | T ≥ t) = a / (a + b + t - 1)

Expected waiting time: E[T] = (a + b - 1) / (a - 1) for a > 1 (Positive infinity for a = 1)

Worked Example & Boundary Behavior

Consider the case where a = 2 and b = 2, and let us evaluate the distribution at trial t = 2:

The Beta function for positive integers is B(a, b) = (a - 1)! (b - 1)! / (a + b - 1)!. For a = 2, b = 2, B(2, 2) = 1! 1! / 3! = 1/6.
For t = 2, B(a + 1, b + t - 1) = B(3, 3) = 2! 2! / 5! = 4 / 120 = 1/30.
The PMF is P(T = 2) = (1/30) / (1/6) = 1/5 = 0.2.
The survival probability is P(T ≥ 2) = B(2, 3) / B(2, 2) = (1/12) / (1/6) = 1/2 = 0.5.
The hazard at t = 2 is h(2) = 2 / (2 + 2 + 2 - 1) = 2/5 = 0.4, and the mean waiting time is (2 + 2 - 1)/(2 - 1) = 3.

Interactive Experiment

Follow these steps in the Beta–Geometric Waiting Time module:

Set Shape a to 2, Shape b to 2, and Trial index to 2.
Observe that probePMF equals 1/5 (0.2), survival equals 1/2 (0.5), and hazard equals 2/5 (0.4).
Compare the decreasing hazard curve with the constant hazard of the plug-in geometric model.
Set Shape a to 1 to observe the boundary regime where P(T ≥ t) decays as O(1/t) and the expected waiting time is Positive infinity.

Interpretation & Limitations

This mixture model requires that P is drawn once per waiting-time run and shared across all trials of that run. Redrawing P independently on every single trial would erase the Bayesian learning effect and collapse the waiting time back to an ordinary Geometric distribution with success probability E[P] = a / (a + b).

References

The following reference provides the analytical formulas and applications of the beta-geometric distribution in reliability and survival analysis:

NIST Dataplot: Beta-Geometric Probability Density Function

Beta–Geometric Waiting Time