Base rates

What you will learn

Explain how a rare state can remain unlikely after evidence that favors it.

Prerequisites: Conditional directions

A large alternative population can generate many signals

Suppose strong hands are uncommon. A player may bet most strong hands and only a small fraction of weak hands, yet weak bets can still outnumber strong bets because there are many more weak hands to begin with.

The relevant counts multiply population sizes by conditional action rates. Comparing action rates alone discards the prior. This is the same base-rate issue encountered in diagnostic tests and classification systems, although a poker action is not a medical test and the model assumptions remain different.

Evidence changes a prior; it does not erase it

A signal that is more likely under strength than weakness raises the probability of strength, provided both states had positive prior mass. Raising a probability from 1% to 8% is meaningful evidence, but the state can remain less likely than its alternative.

The prior must describe the actual population and decision context before this observation. Importing a prior from a different opponent, street or selection process can change the inference. A posterior with many decimal places still inherits errors in those assumptions.

Common mistakes

• Comparing 80% and 20% without their population weights.
• Claiming a posterior increase means the state is now more likely than not.
• Using a prior from a differently selected population.

Keep these ideas

• Weight likelihoods by the sizes of the starting groups.
• Strong evidence can leave a rare event relatively uncommon.
• State the prior’s population and timing.