by mith
Interesting that the geek rating went up by more points than the average rating.
Calculus time!
I am modelling the rating algorithm as follows:
B = (A*V + 5.5*S)/(V + S)
where B = BGG rating, A = Average rating, V = Votes, S = Bayesian/Shill parameter. Given our daily stats, I can track changes in S by rewriting this equation:
S = V*(A - B)/(B - 5.5)
For example, today's value is 777.57 ± 1.91 (where the error is due to uncertainty in A and B of ±0.0005).
Assume V and S are constant. Then for a given change in A, we can calculate the change in B by taking the partial derivative:
δB/δA = V/(V + S)
That means currently, a change in A will result in about 86% of that change in B - in Remy's notation, a 6 point jump in average should result in a 5 point jump in BGG rating,
if the votes remained constant.
We can do the same thing to approximate the change in B due to a change in V:
δB/δV = S*(A - 5.5)/(V + S)^2
This is a very small number (approximately 0.00008 right now); each individual vote isn't worth much, but of course the change in number of votes is also much bigger than the change in average. In Remy's notation, we can say each vote is worth about 1/13th of a point - so 67 new votes today would again correspond to about 5 points in the BGG rating,
if the average remained constant.
So, if S is constant, we would expect B to increase faster than A - it doesn't take many votes to make up that 14% gap. (Of course, in this case the estimate of S also went up by 5, which cause B to only increase by 8 points instead of 10+.)