Re: [computer-go] 9x9 Ratings

[Antti Huima <antti.huima@xxxxxxxxxxxxx>]

Łukasz Lew wrote:

>Hi,
>
>You wrote:
>Then create a probabilistic model that binds these ranks together:
>settle on a probability distribution that maps a difference in two of
>these 20 player-specific ranks to a probability, and take the resulting
>probability into the maximum likelihood equation. (This kind of forces
>the 20 ranks to approach each other)
>
>Could You explain more clearly how can I bind several coefficients
>together through probabilistic model?
>
>Kind regars,
>Lukasz Lew
>  
>
Dear Lukasz,

this is really not complicated. Suppose the 20 ranks are R[1] ... R[20],
ordered so that R[1] is White+H9 and R[20] is Black+H9, i.e. in order of
decreasing skill. Suppose that the numeric values are, say, between 1.0
(9 dan) and 0 (40 kyu).

The idea is to have a function that maps every possible sequence of the
values of the 20 ranks into a probability of that sequence. This is "a
priori" information whose purpose here is to state that it is probable
or common that the ranks are near each other---this is the common or
desired view on go handicaps.

I would proceed so that I would define a probability function that e.g.
yields the "probability that the ranks correspond to a sequence of ranks
possible for a typical human player". I would try a function of the form

P = (1-|R[1]-R[2]|^x) * (1-|R[2]-R[3]|^x) * ... * (1-|R[19]-R[20]|^x)

with a small x (< 1). The idea is that a completely flat sequence (all
ranks are the same) is "surely" one possible for a typical human player
(gets 1), where as a completely random sequence (9 dan, 40 kyu, 4 dan,
19 kyu, ...) is not (gets a low probability). Then this term P for every
player would be just multiplied into the likelihood of the game results.
Note also that I bind here only the successive ranks together,
formulating the idea that maybe the ranks for White H9 and Black H9 can
differ more than the ranks for White H2 and White H3, say.

The net effect should be that it would be impossible to get rankings
where a human player has widely different ranks for different handicaps.
If there would be a such player, then the factor P for that player would
have small numeric value. But that value would be multiplied into the
likelihood, hence the likelihood also would become small, suggesting
that the solution would not be a maximum after all.

I send this through computer-go as I think this fits the thread there.

Yours Truly,

-- 
Antti Huima (Mr.)
Director, Conformiq Tools
mobile: +358 40 528 8667
email: antti.huima@xxxxxxxxxxxxxxxxx

Conformiq Software Ltd.
Stella Terra, Lars Sonckin kaari 16
FIN-02600 Espoo, Finland
tel: +358 10 286 6300
fax: +358 10 286 6309 
http://www.conformiq.com/

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