NCAA Lacrosse Bradley-Terry

[jtwcornell91]

[quote Swampy][quote jtwcornell91][quote Jacob '06]Is the lower probability center for Cornell due to their weaker SOS, or less exposure to OOC teams? (By lower I meant the center of our distribution is only at ~.5 and we have a wider distribution)[/quote]

It's just that in the second plot everything's scaled so that the area under all of the curves is the same.  Since ours is broader, the peak is lower.  The broader distribution means our rating is less precisely determined.  That may be because of the slightly weaker schedule, but it may also just be that our rating is farther from 100, where everybody's prior was peaked at the start of the season.  Note that Duke's distribution is the second-broadest.[/quote]

Wouldn't the spread reflect the total number of games played? The standard deviation is inversely proportional to the sample size, and the curves have a more than passing resemblance to a normal curve.[/quote]

Yeah, the curves are all Gaussians in log(rating).  The maximum likelihood equations are exact, but working with the actual shape of the posterior as a function of 56 variables would be impractical.  (Marginalizing over the other 54 teams' ratings would mean doing a 54-dimensional numerical integration!)  So all of the probabilities actually use the Taylor expansion of log(Posterior) to second order in log(ratings), which is a Gaussian.

Anyway, the standard deviation depends on the number of games played but also on the lopsidedness of the games, according to the maximum-likelihood values for the ratings.  If N(A,B) is the number of games played by A against B, HHWP(A,B) is the predicted head-to-head winning percentage for A vs B according to the ML ratings, and HHWP(A,0) and HHWP(0,A) are the corresponding quantities for team A vs the reference team with a rating of 100, the inverse-sigma-squared matrix is

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invsigsq(A,A) = 2 * HHWP(A,0) * HHWP(0,A) + sum_B [ N(A,B) * HHWP(A,B) * HHWP(B,A) ]
invsigsq(A,B!=A) = - N(A,B) * HHWP(A,B) * HHWP(B,A)

The widths of the Gaussians are the square roots of the diagonal elements of the inverse of this matrix.  So increasing the number of games played does scale up the inverse sigma squared matrix and therefore scale down the relevant sigmas.  But the products of ML HHWPs come into play as well.  Note that

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HHWP(A,B) * HHWP(B,A) is a maximum (1/2*1/2=1/4) when A and B have the same ML rating and goes to zero (0*1=0) when the ratio of the ratings goes to zero or infinity.  So the

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2 * HHWP(A,0) * HHWP(0,A) term will give a higher inverse-sigma-squared and therefore a tighter distribution for a team whose rating is closer to 100.  Also, the

HHWP(A,B) * HHWP(B,A)

inside the sum means that a game does more to tighten a team's distribution when the two teams playing are of similar strengths.  (So for stronger teams, a strong schedule is a way to more precisely nail down the ratings, which makes intuitive sense.  I learn more about Maryland's strength as a team from a game against Georgetown than from a game against Bellarmine.)

[RichH]

[quote jtwcornell91]but working with the actual shape of the posterior as a function of 56 variables would be impractical.[/quote]

Don't I know it!

*rimshot*

[Hillel Hoffmann]

John, you rule.