[computer-go] ELO Ratings of move pattern

[Rémi Coulom]

Jason House wrote:
>
>
> On Dec 6, 2007 11:38 AM, Rémi Coulom <Remi.Coulom at univ-lille3.fr 
> <mailto:Remi.Coulom at univ-lille3.fr>> wrote:
>
>     Jason House wrote:
>     >
>     > This may serve as a good test of if there is enough data to assign
>     > values to the patterns.
>
>     I did not mention this in my paper, but you can rather easily
>     estimate
>     uncertainty margins around Elo values. This involves computing the
>     second-order derivative of the probability distribution with
>     respect to
>     log(gamma). Since the distribution has a shape that looks very
>     much like
>     a Gaussian, the second-order derivative at the maximum is a good
>     estimation of -1/sigma². That is how I compute confidence intervals in
>     bayeselo.
>
>
> What do you mean by the probability distribution with respect to 
> log(gamma)?  Do you mean a plot of the prediction rate with only the 
> gamma of interest varying? 

No the prediction rate, but the probability of the training data. More 
precisely, the logarithm of that probability.

If you have P(x)=A*exp(-x²/2sigma²), then log(P(x))=log(A)-x²/2sigma², 
and d²(log(P(x)))/dx²=-1/sigma². This means that, for a Gaussian 
probability distribution, the second-order derivative directly gives the 
variance. For distributions that look similar to a Gaussian, the 
second-order derivative is a good approximation.

Rémi