Evidence from a check
Use the likelihood of the action actually observed, including an absent bet.
10 minutes · intermediate · Original Poker4Quant lessonWhat you will learn
Update from a check by complementing the betting likelihood within each state.
Prerequisites: Bayes with a frequency table
No bet is an observation too
Suppose the only available modeled actions are bet and check. Within each state, their probabilities sum to one. Thus P(check|S)=1−P(bet|S), and likewise for weak hands. These are complements within state rows, not complements of the final posterior.
If a player also had other modeled actions, such as folding to a bet, “not bet” could include more than a check. Be explicit about the action set before interpreting an absence.
Do not complement the wrong quantity
P(S|check) is generally not 1−P(S|bet). Those posteriors have different conditioning populations. The complement of P(S|bet) is P(weak|bet), because the bet evidence stays fixed while the state changes.
A check may favor weakness if strong hands bet more often, but it need not prove weakness. Players can check strong hands in the model. The amount of evidence depends on how differently the two state groups check, weighted by their priors.
Worked example
Strong hands have prior 25%. Bet rates are 80% when strong and 40% when weak. After a check, what is P(strong)?
- Check likelihoods are 20% when strong and 60% when weak.
- In 100 modeled hands, strong checks contribute 25×0.20=5.
- Weak checks contribute 75×0.60=45.
- The posterior is 5/(5+45).
P(strong|check)=10%. This is not obtained by subtracting P(strong|bet) from one.
Common mistakes
- Using betting likelihoods after observing a check.
- Taking one minus a posterior with a different conditioning event.
- Treating a check as certainty that a hand is weak.
Keep these ideas
- Complement likelihoods within each state when actions are exhaustive.
- Keep the observed evidence fixed when taking a posterior complement.
- A check can change a range without uniquely identifying a hand.