The incomplete beta function is defined by

It is called **incomplete** because the limit of integration doesn’t necessarily run all the way up to 1. When *z* = 1 the incomplete beta function is simply the beta function

discussed in the previous post [1].

The incomplete beta function is proportional to the CDF of a beta random variable, and the proportionality constant is 1/*B*(*x*, *y*). That is, if *X* is a beta(*a*, *b*) random variable, then

The right-hand side of the equation above is often called the **regularized** incomplete beta function. This is what an analyst would call it. A statistician would call it the beta distribution function.

The previous post gave simple bounds on the (complete) beta function. This post gives bounds on the incomplete beta function. These bounds are **more complicated** but also** more useful**. The incomplete beta function is more difficult to work with than the beta function, and so upper and lower bounds are more appreciated.

One application of these bounds would be to **speed up software** that evaluates beta probabilities. If you only need to know whether a probability is above or below some threshold, you could first compute an upper or lower bound. Maybe computing the exact probability is unnecessary because you can decide from your bounds whether the probability is above or below your threshold. If not, then you compute the exact probability. This is less work if your bounds are usually sufficient, and more work if they’re usually not. In the former case this could result in a significant speed up because the incomplete beta function is relatively expensive to evaluate.

Cerone, cited in the previous post, gives two lower bounds, *L*_{1} and *L*_{2}, and two upper bounds, *U*_{1} and *U*_{2}, for *B*(*x*, *y*, *z*). Therefore the maximum of *L*_{1} and *L*_{2} is a lower bound and the minimum ofÂ *U*_{1} and *U*_{2} is an upper bound. The bounds are complicated, but easier to work with than the incomplete beta function.

where

## Related posts

[1] Incidentally, Mathematica overloads the `Beta`

function just as we overload *B*. Mathematica’s `Beta[x, y]`

corresponds to our *B*(*x*, *y*), but the three-argument version of `Beta`

, the incomplete beta function, puts the limit of integration *z* first. That is, our *B*(x, *y*, *z*) corresponds to `Beta[z, x, y]`

in Mathematica.