I have no training in Bayesian data analysis, so I can't wrap my head around how to start solving the following problem and am hoping you can help:
I am using linear regression to forecast the net scores (home - visitor) of (American) pro-football games from differences in team-strength scores (home - visitor). Those strength scores fall on a 0-100 scale, and they represent the percent chance that the team in question would beat another team selected at random from the 31 others in the league. The differences between those strength scores and the net game scores are both normally distributed.
Right now, I am using team-strength scores that are fixed for the entire season in a mixed-effects model that also includes random intercepts for each team as the home team. The strength scores are fixed because they come from a preseason survey. I would like to see if I can make the predictions more accurate by using Bayesian updating to allow that team-strength score to vary over the course of the season, as we learn more about how teams are performing relative to preseason expectations.
The single piece of information that strikes me as most useful in that regard is the cumulative sum of each team's prediction errors --- in other words, the cumulative sum of the differences between the team's predicted game performance (based on the preseason strength scores and where each game is played) and its actual game performance.
How might I go about doing that? In R, I have gotten as far as computing those cumulative errors, which turn out to be normally distributed for the season with a mean of ~0 and sd of ~50. I have tinkered with algebraic ways to adjust the strength scores as a function of that cumulative error. The forecasts based on those algebraic adjustments are slightly more accurate, but the approach seems clunky, and I'd like to use this problem as an opportunity to learn about Bayesian updating if I can. Any suggestions on how to do that in the context of this problem --- and, ideally, in R --- would be much appreciated.