I am working on an extension of the Glicko2 rating system to use in predicting the outcome of sport/game contests that uses the actual head-to-head results of previous meetings of competitors to provide more information in calculating the probability of each side winning a particular game. What I am trying to do is say that the ratings provided by Glicko2 give the prior probabilities of each side winning, and use the head-to-head results as the Bayesian updating data. That is, I am looking for the probability of a side winning given the head-to-head results, using the probability from the rating system as my prior. In order to use Bayes' Theorem, however, I need to know things that I'm not really sure how to figure out even with access to all the data I have. Specifically, I need to know the probability that the head-to-head data would be observed "under the hypothesis" that a certain side wins the next match. How exactly to do this is not particularly simple. I was initially unsure of even anything about how to calculate these quantities and asked a question at a general-information intellectual message board, and no one there could help me but suggested I try here. My thread there is the top google result (at least for me) for "updating rating systems probabilities using head-to-head results", and all other results are not even close to envisioning the situation that I have constructed for myself.
I'm not even sure what the "probability of observing the data given the hypothesis" even means in this situation. In other Bayes' Theorem problems I've looked at, those probabilities are given or very simple to find. I thought I had some sort of answer, but after coding it up noticed that it was returning absurd results, specifically that a side with a 75% win chance as calculated from ratings and a 60% win rate in prior contests against the opposing side would win less than 60% of the time. Clearly there is something wrong with my calculations there. I was modeling the probabilities of the data given each side winning as though the data were being drawn from a binomial distribution with probability as implied by what the two sides' ratings would be updated to given that side winning. I also tried taking the binomial distribution's probabilities from ratings at the time of the previous contests, but the absurd results were still there, and it was far less obvious how to modify the binomial probability based on each side winning.
Where is the mistake in my approach here? How should I be finding the conditional probabilities that are reversed from the ones that I'm actually trying to calculate? Is this even not actually a Bayes' Theorem application because such conditional probabilities are not obvious, and I'm better off looking for the probability I'm most interested in a more direct manner?