normal vs logistic curve for elo model

[Daniel Shawul]

Does anyone know how to derive the bayeselo equations for home field advantage and draws ? I think I went well upto the minorizing function ok but lost afterwards. I will post what I did so far incase someone has any idea how to finish it.

I considred home losses, wins and draws only. Since the summation for both i and j is over all the indices the away WDLs will be the same as home WDLs for the other side.

Code: Select all

P(WinAtHome) = thetaH * gammaI / (thetaH * gammaI + thetaD * gammaJ)
P(LossAtHome) = gammaJ / (thetaH * thetaD * gammaI + gammaJ)
P(DrawAtHome) = (thetaD^2 - 1) P(WinAtHome) P(LossAtHome)

Likelihood

Code: Select all

L(gamma) = Product( P(WAH) ^nwah * P(LAH) ^ nlah * P(DAH) ^ ndah)
lnL(gamma) = Sum over i,j { (nwah + ndah) P(WAH) + (nlah + ndah) P(LAH) + (ndah) * ln(thetaD^2 - 1) }

How to go from here? I applied the minorizing function but got some extra terms even in the example given there which only considered draws. I guess those terms are just dropped since we just want to maximize anyway. But I want to understand how.
thanks.