[Feature Request]: Add betaincinv and gammaincinv

[amyoshino]

Before

No response

After

No response

Context for the issue:

I would like to initiate this matter to formally request the inclusion of the following functions:

• betaincinv: Inverse of the regularized incomplete beta function, or quantile function for the beta distribution.
• gammaincinv: Inverse to the regularized lower incomplete gamma function.

The implementation of these functions will allow us to implement quantile/ICDF functions for distributions like Beta, Gamma, ChiSquared and StudentT distributions, being fundamental to the issue: pymc-devs/pymc#6845

Some references I think might be useful:

• betaincinv function in SciPy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammaincinv.html
• gammaincinv function in SciPy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betaincinv.html
• CEPHES implementation of betaincinv: https://github.com/scipy/scipy/blob/main/scipy/special/cephes/incbi.c
• CEPHES implementation of gammaincinv: https://github.com/scipy/scipy/blob/main/scipy/special/cephes/igami.c
• Very neat implementation of betaincinv for TensorFlow probability: The inverse of the regularized incomplete beta function tensorflow/probability#1581

[ricardoV94]

That would be really nice. Would you like to take a stab at it?

As a first approach I would try to create symbolic expressions instead of creating new Ops/wrapping C code. This way it will more easily work on other backends later down the road.

You already tooks a look, but again, the scalar gammainc and betainc grads are good examples.

[amyoshino]

I'd be more than happy to give it a try. I will require some guidance as I work on it, and I'll keep you posted with a pull request as soon as I make some headway. Your support is greatly appreciated!

[ricardoV94]

Happy to help and thanks in advance!