Bayesian Statistics in Poker

Credible intervals

Most statistics in poker take the form of a percentage. For example, the percentage of hands that a player raises before the flop (PFR) or the percentage of hands that a player sees the showdown (WTSD). Most software provides the following single value:

\mbox\{sample mean\} = \frac\{\mbox\{observed instances\}\}\{\mbox\{observed opportunities\}\}

When the number of observed opportunities is large, the value is a good estimate of the player's overall behavior. Unfortunately, when few opportunities have been observed the sample mean is not a good estimate. For example, suppose a new player sits down at your table and folds the first two hands on the flop. How often do we estimate how often this player sees the showdown when they've seen the flop? The sample mean is 0%0 observed instances / 2 observed opportunities = 0%, but it's implausible that this player never sees the showdown.

Instead of displaying a single value, Poker Sleuth shows you a range of percentages that the player could credibly have. From the example above, Poker Sleuth might display 36±35, meaning it estimates the true percentage as between 36-35=1 and 36+35=71. If the player saw the showdown on 5 of their next 10 flops, Poker Sleuth might revise the credible interval to 44±25, meaning it estimates the true percentage as between 44-25=19 and 44+25=69.

Bayesian statistics

Poker Sleuth takes statistics to the next level, by developing a profile of typical player behavior based on other players you've observed. These profiles are incorporated into Poker Sleuth's computation of credible intervals in a mathematically rigorous way. For example, Poker Sleuth may note that nearly all players see the showdown between 29% and 65% of the time when they see the flop. When a new player sits down, Poker Sleuth will show this statistic as 47±18. As you observe the player, Poker Sleuth will revise the estimate, slowly converging on the player's true behavior.

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