Bayes with a frequency table

What you will learn

Construct a complete two-state table and normalize the observed column.

Prerequisites: Base rates

Fill both states and both actions

Let the prior strength probability be p. If the bet likelihood when strong is a and when weak is b, the four joint probabilities are pa, p(1−a), (1−p)b and (1−p)(1−b). They cover strong/weak crossed with bet/check.

The probability of a bet is pa+(1−p)b. Conditioning on that column gives P(S|B)=pa/[pa+(1−p)b]. A frequency table makes it easier to audit that every observed case appears once.

The evidence denominator must be possible

If both states assign zero probability to the observed action, the model says it cannot happen. Dividing zero by zero does not produce a meaningful posterior. The observation then challenges the model rather than licensing an arbitrary answer.

Frequency examples may use a convenient hypothetical population to make the counts integers. Those counts explain the model; they are not claimed empirical observations of an opponent. Distinguish an illustrative table from data collected in a real study.

Common mistakes

• Normalizing by only one state’s contribution to the evidence.
• Leaving checks out and accidentally treating bets as the whole experiment.
• Assigning a posterior to a zero-probability conditioning event.

Keep these ideas

• Build joint state/action cells.
• Add the full observed column for the denominator.
• Impossible evidence requires revisiting the model.