betai

Betai is a common notation in numerical libraries for the incomplete beta function, a special function with strong ties to the beta distribution. In practice, betai(a, b, x) may refer to either the incomplete beta function B_x(a, b) or its regularized form I_x(a, b), depending on the software. The incomplete beta function is foundational in probability and statistics, while the regularized version is widely used as a CDF for the Beta distribution. The safe mathematical definitions are:

- B_x(a, b) = ∫_0^x t^{a-1} (1−t)^{b−1} dt, for a > 0, b > 0, 0 ≤ x ≤ 1.

- B(a, b) = ∫_0^1 t^{a-1} (1−t)^{b−1} dt = Γ(a)Γ(b)/Γ(a+b).

- I_x(a, b) = B_x(a, b) / B(a, b), the regularized incomplete beta function.

Key properties include the symmetry I_x(a, b) = 1 − I_{1−x}(b, a) and the derivative d/dx I_x(a, b) = x^{a−1} (1−x)^{b−1} / B(a, b). Special values occur at x = 0 or 1, and when a or b is a positive integer, simplifications via summations are possible.

Computation typically relies on stable representations such as continued fractions for the ratio I_x(a, b) when x is near 1, and series expansions or recurrence relations for other regimes. Numerical implementations emphasize stability for extreme parameter values and large a, b, often using logarithmic forms of Gamma/Beta functions to avoid overflow.

Applications of betai (and I_x) include computing the CDF of the Beta distribution, Bayesian inference with Beta priors, and various statistical tests and order-statistic calculations. When encountering betai in code, consult the specific library documentation to confirm whether the function returns B_x(a, b) or I_x(a, b).

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