I have a general question about Bayesian inference which may help me solve a problem I have. It is best to illustrate this with an example. Inspired from this great post by AllenDowney:
Let's say I want to get an estimate for the number of goals scored by a team in a football match. I model this as a Poisson process with parameter $\lambda$. On $\lambda$ I put a Gamma distribution as the prior, with parameter $\alpha$. Let's say I use a prior value of 1.4 for $\alpha$ (avg number of goals scored by a team).
In the blog post, we are computing the posterior after a match has been played. So, given a game where 4 goals were scored by a team, we compute the posterior for that team which is now shifted to the right.
What would we do if we wanted to update the estimate in real-time? So instead of computing the posterior after the game has been played, we do this every 5 minutes until we hit 90 minutes. So I am interested in getting a posterior for 90 minutes given the data after 5 minutes, 10 minutes, 15 minutes, etc.. I can think of two ways of doing this. Please help me understand what method, if any, makes sense:
We propagate the data we have to the expected number of goals in 90 minutes after $x$ minutes have passed. So, if after 30 minutes a team scored 1 time, we propagate that to (90/30)*1=3 goals and use this as input to compute the posterior using the same process as in the blog post. Issue: I feel like this is not correct, because what if a team scores in the first 5 minutes? Doing this would mean we expect the team to score 18 (!) times in a match. The first posterior calculation after 5 minutes would be complete trash. Although this should eventually converge to a realistic value as we do more updates?
We don't use a value of 1.4 for $\alpha$ because that's based on a 90 minute match. Instead, we use alpha = 1.4 / (90/5) = 0.08 to scale it to a 5-minute prior. So, on average, a team scores 0.08 times every 5 minutes. We now do the posterior calculation every 5 minutes instead of every 90 minutes. Issue: I don't understand how we now get a prediction for the 90 minute posterior because the posterior we calculate every time will be based on 5 minutes. Also, how do we link the second posterior (for minutes 5-10) to the first (minutes 0-5)?
Perhaps I am missing something basic here. I really want to understand Bayesian inference better but feel like I am not completely getting it. Thanks!