Quote:
Originally Posted by AGPapa
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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Following up on this, the attached image is snipped from the
Wikipedia Page on MMSE Estimation about half way down.
In this equation, x is the parameter you're trying to estimate (like O), z is the noise (like N), xhat is your estimated parameters, xbar is the expected mean, Cx is the covariance matrix of x, and Cz is the covariance matrix of the noise.
For example, in my MMSE equation for the OPRs, xbar is just Oave (but you could have it be a vector with team specific expectations). A* xbar is just the average match outcome which I have as 3*Oave (but again, if you expect xbar to be team specific then that would cause non-constant match mean scores). Cz is just sig2n * I and Cx is just sig2o * I. I plugged these in and simplified the equations. But if things are more complicated (like with EPR), then you just plug in whatever complicated assumptions you have and go from there.
It would be neat if we could study the best predictor of a team's OPR at championships from the OPR they had in their last regional before championships using data from previous years. We'd probably come up with a mean and variance of this best predictor, and then we could plug these in and have some expectations for what championships would look like even before the first match was played. Then as matches are played, the values update to include the new information using the MMSE equation with changing A and Mo.