I'm using a beta binomial updating model for a piece of code that I am writing. The software is real time updating - meaning that data is continually being gathered and after N data points are gathered, the bayesian model is updated using the N data points.
Under this logic, I am using the posterior output as my prior for the next iteration. My problem is that over billions/trillions/maybe more of iterations, the bayesian beta parameters (alpha and beta) will grow very large. I am worried that eventually the parameters will become so large that they will cause an integer overflow in memory.
So my question is twofold -
Is it reasonable to be worried about this integer overflow. I understand that $2^{32}$ is an extremely large number, but I'm building this software for an internet service that will be running 24/7, 365 days a year and I don't want it to crash. For example if I was updating it with 1,000,000 data points a day then the model would only last ~4000 days before an integer overflow.
Is it possible to transform a Beta(x,y) r.v., where x and y are extremely large, to a Beta(x*,y*) r.v. where x* and y* are relatively smaller? The transformed Beta doesn't have to be exact, just similar.