You seem like you have a solid statistics background, so I'm going to use some mathematical notation in my explanation, but feel free to ask me to explain any of it if the meaning is unclear.
Quote:
Originally Posted by poincaraux
I obviously have no idea what you're actually doing, but the first thing I thought when I looked at the above graph was "oh, that looks like the CDF of a Gaussian, but shouldn't he be using a Student's t-distribution b/c of small sample size?"
Poker Sleuth uses Beta distributions (which are similar enough to Gaussian distributions that you can approximate a Beta distribution with a Gaussian), but have two key advantages when estimating probabilities:
- Pr(x < 0) = Pr(x > 0) = 0%
- Variables cancel out nicely when put through Bayes' Theorem
Quote:
Originally Posted by poincaraux
- Are your ranges just 95% confidence intervals?
Technically, the ranges are 95% Bayesian credible intervals, which are similar to confidence intervals but not the same thing (I wish I could find a good article that explained the difference crisply, but I can't at the moment. The Wikipedia entries are OK) In particular, confidence intervals are prohibited from incorporating any extra information beyond the observations.
(In the Poker Sleuth Options window, you can actually adjust the "95%" to be whatever you want)
Quote:
Originally Posted by poincaraux
- What sort of Bayesian analysis are you doing? Is it things like "we don't have enough hands to say anything directly about river c/r ranges, but given his flop aggression, etc., etc., a 95% CI would be XX-YY"? I'd love to see some data about which statistics correlate with each other, which don't, etc.
For some statistic S on a player P, Poker Sleuth currently does the following:
- Create a model M for the distribution of the statistic across all players
- Apply Bayes' Theorem to create a PDF function f(x) = Pr(S=x given M and the observations on this player) and the similar CDF function F(x)
- For a 95% credible interval, give F(2.5%) to F(97.5%) as the range
If we have no observations on a player, then F(x) is the same as the distributions across all players.
I hope that makes sense at least at a very high level.
A future version of Poker Sleuth may take into account how different statistics correlate with one another. The hardest part, as you may imagine, is determining how statistics correlate. ;-)
The preflop analysis tool ("Poker SHIP") that's part of Poker Sleuth does do some more sophisticated stuff using some basic correlations. For example, it's smart enough to know that if a player raises frequently with TJs from early position, they probably also raise with QJs from late position.