141. Bayes' Rule

141.1. Beyes’ Rule

Start with the multiplication rule in both directions:

Set them equal:

Solve for $𝑃\left(𝐴|𝐵\right)$:

So Bayes’ Rule is the multiplication rule rearranged.

Start: Definition of conditional probability

1. Numerator: $𝑃\left({𝐴}_{𝑖}\cap 𝐵\right)$ (Multiplication Rule)

Using the multiplication rule:

This expresses the joint probability as prior $\times$ likelihood

So the numerator comes directly from multiplying:

• prior $𝑃\left({𝐴}_{𝑖}\right)$
• likelihood $𝑃\left(𝐵|{𝐴}_{𝑖}\right)$

2. Denominator: $𝑃\left(𝐵\right)$ (Total Probability Theorem)

If ${𝐴}_1,\dots ,{𝐴}_{𝑛}$ form a partition, then the total probability theorem gives:

This decomposes the probability of the “evidence” $𝐵$ into all possible scenarios ${𝐴}_{𝑗}$

3. Put numerator & denominator together

Substitute both results into the definition:

Interpretation

• Numerator: how strongly scenario ${𝐴}_{𝑖}$ explains the evidence $𝐵$ (likelihood $\times$ prior)
• Denominator: total probability of seeing the evidence at all (sum of all scenario contributions)

So Bayes’ Rule is literally: