Estimating Phil Galfond's win rate against VeniVidi1993
02-02-2020
, 02:24 AM
Feb 1, 2020. Phil is down €452,523.65 after 6146 hands; I'm using std dev 183bb/100.
I would like to graph Phil Galfond's win rate, as a probability density function, given the current results.
---
Galfond EV (bb/100)= -36.81448503
Std Dev after 6146 hands =23.34290012
My first thought was to simply graph a normal distribution, with mean -37 and std dev 23.34.
This looks fine at first, but there are problems. This distribution assumes that it is just as likely that Phil has a win rate of -60bb/100 as he is to have a win rate of -13.5bb/100. (-36.8 +/- 23.34).
So I came up with another strategy that would assign more weight to values towards zero, and less weight to the extremities.
The "weighted probability function" works as follows:
Left = probability of EV assuming a true win rate of -37bb/100.
Right = probability of EV assuming a true win rate of 0bb/100.
The middle is the weighted probability function, obtained by multiplying and normalizing the "left" and "right" functions.
Is this approach valid?
I would like to graph Phil Galfond's win rate, as a probability density function, given the current results.
---
Galfond EV (bb/100)= -36.81448503
Std Dev after 6146 hands =23.34290012
My first thought was to simply graph a normal distribution, with mean -37 and std dev 23.34.
This looks fine at first, but there are problems. This distribution assumes that it is just as likely that Phil has a win rate of -60bb/100 as he is to have a win rate of -13.5bb/100. (-36.8 +/- 23.34).
So I came up with another strategy that would assign more weight to values towards zero, and less weight to the extremities.
The "weighted probability function" works as follows:
Left = probability of EV assuming a true win rate of -37bb/100.
Right = probability of EV assuming a true win rate of 0bb/100.
The middle is the weighted probability function, obtained by multiplying and normalizing the "left" and "right" functions.
Is this approach valid?
02-19-2020
, 09:20 AM
Read up on "Conjugate priors" (but there are probably much better resources to learn this from than the Wiki page...).
Specifically look at the "Posterior hyperparameters" for the "Normal with known variance σ^2" in the table.
If you assume a prior of distribution of N(0, 23.34) and then update with a single observation (ie: n = 1 in the equation above) of N(-37, 23.34) you get exactly what you've drawn.
The problem is that:
1. You really should choose your prior distribution beforehand.
2. You should do the update in terms of n observations (where n is per hand or per block of 100 hands, etc).
If you do it properly then you should see the middle (posterior) distribution slowly move towards the observed (likelihood) distribution and away from the prior distribution, as you get more observed hands (which is what you would expect; rather than it being right in the middle of the two always...).
Juk
Specifically look at the "Posterior hyperparameters" for the "Normal with known variance σ^2" in the table.
If you assume a prior of distribution of N(0, 23.34) and then update with a single observation (ie: n = 1 in the equation above) of N(-37, 23.34) you get exactly what you've drawn.
The problem is that:
1. You really should choose your prior distribution beforehand.
2. You should do the update in terms of n observations (where n is per hand or per block of 100 hands, etc).
If you do it properly then you should see the middle (posterior) distribution slowly move towards the observed (likelihood) distribution and away from the prior distribution, as you get more observed hands (which is what you would expect; rather than it being right in the middle of the two always...).
Juk
Last edited by jukofyork; 02-19-2020 at 09:27 AM.