On Dec 5, 2007 4:44 AM, Lars <[EMAIL PROTECTED]> wrote:
> I have some questions concernig this paper of Remi:
> http://remi.coulom.free.fr/Amsterdam2007/MMGoPatterns.pdf
>
> @Remi: How many iterations you had used?
>
> Anyone of you have similar or other experiences with the algorithm?
I seem to have more time to think than to code lately. I believe I've
derived an alternate update method. I'd be curious how quickly it
converges.
Using Minorization-Maximization, Remi derived
new gamma = wins/sum(C/E)
Using Newton-Raphson method, I derived
fractional change = (estimation error) * (fractional uncertainty)^2
new gamma = gamma * (1 + fractional change)
Where
estimation error = (wins-expected wins)
fractional uncertainty = sigma/gamma
fractional change = new gamma/gamma - 1
Changing gamma based on both the estimation error and the uncertainty that
gamma is correct makes intuitive sense to me. Sigma matches remi's
definition. Because strengths are multiplied together, fractional
uncertainty makes more sense. I'm avoiding the difficult issue of
explaining why, but I do point people to consider two things:
1. If gamma=gamma+sigma, what does that do to team strength?
(It multiplies the team strength by 1 + fractional uncertainty)
2. Look at the similar definitions of fractional uncertainty
and fractional change.
A less intuitive form is:
fractional change = (wins-expected wins) / (expected wins^2 - wins)
Using Remi's notation, expected wins = gamma*sum(C/E)
Regardless of if someone else plays with the alternate equation, I'll be
playing with both after the holidays. I'd also be curious what thoughts
people have about this.
To derive this, I applied the Newton-Raphson method to find where the
partial derivative of the log-evidence is zero (AKA, where log-evidence is
maximized). If L = log evidence, and *L = 1st partial derivative of L with
respect to gamma and **L = 2nd partial derivative of L with respect to
gamma, then new gamma - gamma = *L/**L.
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