Numba Support For Big Integers?

I have a factorial lookup table that contains the first 30 integer factorials. This table is used in a function that is compiled with numba.njit. The issue is, above 20!, the numbe

Solution 1:

There may be some way to handle large integers in Numba, but its not a method that I'm aware of.

But, since we know that you're trying to hand-code the evaluation of the Beta distribution in Numba, I have some other suggestions.

First though, we must be careful with our language so we don't confuse the Beta distribution and the Beta function.

What I'd actually recommend is moving all your computations on to the log scale. That is, instead of computing the pdf of the Beta distribution you'd compute the log of the pdf of the Beta distribution.

This trick is commonly used in statistical computing as the log of the pdf is more numerically stable than the pdf. The Stan project, for example, works exclusively to allow the computation of the log posterior density.

From your post history I also know that you're interested in MCMC; it is also common practice to use log pdfs to perform MCMC. In the case of MCMC, instead of having the posterior proportional to the prior times the likelihood, on the log scale you would have the log-posterior proportional to the log-prior plus the log-likelihood.

I'd recommend you use log distributions as this allows you to avoid having to ever compute $\Gamma(n)$ for large n, which is prone to integer overflow. Instead, you compute $\log(\Gamma(n))$. But don't you need to compute $\Gamma(n)$ to compute $\log(\Gamma(n))$? Actually, no. You can take a look at the scipy.special function gammaln which avoids having to compute $\Gamma(n)$ at all. One way forward then would be to look at the source code in scipy.special.gammaln and make your own numba implementation from this.

In your comment you also mention using Spouge's Approximation to approximate the Gamma function. I've not used Spouge's approximation before, but I have had success with Stirling's approximation. If you want to use one of these approximations, working on the log scale you would now take the log of the approximation. You'll want to use the rules of logs to rewrite these approximations.

With all the above considered, what I'd recommend is moving computations from the pdf to the log of the pdf. To compute the log pdf of the Beta distribution I'd make use of this approximation of the Beta function. Using the rules of logs to rewrite this approximation and the Beta pdf. You could then implement this is Numba without having to worry about integer overflow.

Edit

Apologies, I'm not sure how to format maths on stack overflow.