How can I prevent Overflow for the Gamma function?

Question

I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation

$p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$

I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).

Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood?

Answer 1

You suggest in comments that you don't know how the log-gamma function will work with the likelihood.

You need some basic facts:

1) log of a product of terms is the sum of the logs ($\log(ab) = \log(a) + \log(b)$)

2) log of a reciprocal is the negative of the log ($\log(1/c) = -\log(c)$)

3) The relationship between factorials and Gamma functions ($x! = \Gamma(x+1)$)

Using those three facts you just take the log inside that product (making it a sum of log-terms via rule 1), and apply rules 1,2, and 3 repeatedly until you have nothing but sums and differences of log-gamma functions inside the summation.

(This is not really a statistical issue at all, but a mathematical one, and the problem being solved is a computational one.)

If Python also has a function for the log of a binomial coefficient, you may be able to shave some time off your calculation.