I have a group a baseball players and I have to calculate the expected beta distribution of each player.
Groups
Group 1: Hits: 30, Misses: 471 -> $Beta_1(30, 471)$ = a/(a+b) = 5.99%
Group 2: Hits: 31, Misses: 385 -> $Beta_2(31, 385)$ = a/(a+b) = 7.45%
Note: Group-level data have variable timeframe, we consider the minimum timeframe with at least 30 hits for stats relevance. So If a group of players makes 30 hits in a week, we consider only last 7 days. 30 hits is good so we can trust the hits/miss ratio.
Now I have to estimate the expected beta of a single player within its group. For player data, we consider all available data, so it might be 1 year or a few days. This mean that a player hits could exceed the group hits.
I've read this answer to calculate the new expected beta. Example player A of group 1.
Player A-1 Hits: 176 | Miss: 1734 -> Beta (30+176, 471+1734) = (a+176)/(a+176 + b+1734) = 8.54%
Player B-1 Hits: 1 | Miss: 55 -> Beta (30+1, 471+55) = (a+1)/(a+1 + b+55) = 5.55%
This doesn't look good, because this player B of group A will have a high expected mean (5.55%) despite a very poor performance (1 hits and 55 miss, $success\_rate=1.17\%$). What do you suggest doing?
I've thought calculating a,b in this way because I have group level data.
$alpha = 1.0$
$beta = (1.0/success\_rate_{group}) - 1.0 $
where $ success\_rate_{group} = hits / (hits+miss) $
Does it make sense?
