Proof of Bayes Rule

  Here we prove Bayes Rule from probability calculus. P(A) denotes the probability that event A will occur. Belief measure obeys the three basic axioms of probability theory:

$$0 \leq P(A) \leq 1$$

$$P(sure\;proposition) = 1$$

$$P(A\;or\;B) = P(A) + P(B)$$

The proof derives from the multiplication rule of probability theory which is:

$$P(AB) = P(A \mid B)P(B)$$

Then:

$$P(H \mid e) = \frac{P(He)}{P(e)}$$

By swapping He and using the multiplication rule again:

$$= \frac{P(e \mid H) P(H)}{P(e)}$$

qed.