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Originally Posted by AGPapa
Thanks Will, the fact that even choosing a bad value for Var(N) gives decent results alleviates a lot of my concerns about searching for Var(N) and Var(D) after the fact.
It’s also impressive at how much better MMSE techniques are for when an event is underway and not a lot of matches have been played. This is helpful for predicting the outcome of the second day of matches (and thus seeing who is likely to be a captain).
Is this behavior typical for all stats or just OPR? Could you run a similar plot for sCPR? (since that stat seems to do slightly better than OPR).
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Plots are attached for 2 runs: one with the true Var(D)=0.1 & Var(N)=3, and one for Var(D)=0.0 & Var(N)=3. (these are relative to Var(O)).
The top row is with the estimated Var(D)=0 and the estimated Var(N)=0, 1, 3, and 10 as before. So the top left corner is regular OPRs and the top row is MMSE OPRs.
The middle row is the same but with Var(D) estimated at 0.1 and the bottom row is the same with with V(D) estimated at 0.2.
Note that with Var(D)>0 and Var(N)=0, the results are always the same, the "vanilla" LS sCPR. That's shown in the left middle. Even worse than the OPR, it doesn't really start having values until the rank of the matrix is 2*#teams-1 which is at the 7th match per team. It is VERY overfit which is why it starts noisy and stays noisy.
The bottom left shows the plots of the percentage of the combined O+D (or O in the case when D=0) left after prediction. It's saturated to be no worse than 100%, though the LS OPR and sCPR are worse than 100% when the number of matches is small, meaning that the prediction error has more variance than the original set of parameters (!). The black curve is the LS OPR and the red curve is the LS sCPR which is so overfit that it's worse than nothing until the very last match is played.
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Additionally, how would you implement the techniques described in the “Advanced MMSE Estimation” section? What would you change in the pseudocode to, for instance, change a team’s apriori Oi?
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The regular MMSE equation is like the equation shown in Appendix B for EPR, except that 1/sig2C I there is the inverse of the correlation matrix of the expected parameters. If the expected means change, you don't add Oave at the end but a vector of your expected means. If the expected variances change for the noise or the parameters, then you change the covariance matrices.
Basically, there's a general equation for arbitrary expected mean vectors and covariance matrices of both the parameters and the noise, so you can run the estimation algorithm given any set of expected mean vectors and covariance matrices.