Uncertain opponent frequencies

What you will learn

Compare posteriors across plausible behavioral likelihoods without inventing empirical precision.

Prerequisites: Evidence from a check · Sensitivity and model error

A likelihood input is often an estimate

The earlier exercises state betting rates as known model inputs. In practice, rates may come from limited, selected observations. Three bets in four observed cases does not establish that a stable probability is exactly 75%, especially if the situations differ.

Observed showdowns are also a selected sample: you may not learn the cards behind folds or uncontested bets. A frequency estimated only from revealed hands need not represent the full population of decisions. More data from the wrong selection process can preserve a bias.

Vary the uncertain mechanism

Hold the prior and strong-hand bet rate fixed while varying the weak-hand bet rate. A higher weak-hand rate makes an observed bet less selective, reducing the posterior probability of strength. This is a sensitivity statement within the two-state model.

A full Bayesian treatment could place distributions over uncertain rates and integrate their joint predictive probabilities. Substituting one average rate is not generally identical to averaging the resulting posteriors. Do not present an arbitrary scenario range as a formally estimated credible interval.

Common mistakes

• Treating a handful of observations as an exact stable probability.
• Ignoring the selection process that reveals opponents’ hands.
• Averaging posteriors and calling it the posterior from averaged inputs.

Keep these ideas

• Distinguish inputs given by a problem from estimates supported by data.
• Selection can alter observed frequencies.
• Use sensitivity to expose uncertain likelihood assumptions.