2023-11-29

Inverse Fallacy

Recall the proof of the inverse probability rule, a trivial reformulation of the product rule. For any nonzero probability event $\mathrm{H}$ and D:

• $P(H$ and $D)=P(D$ and $H)$ (product of probabilities commute)
• $P(H \mid D) \times P(D)=P(D \mid H) \times P(H)$ (applying product rule to both sides)
• $P(H \mid D)=P(D \mid H) \times P(H) / P(D)$ (the inverse probability rule)

Note that joint probabilities, the product of two probabilities, commute-i.e., the order of the individual probabilities does not change the result of their product: $$ P(H \text { and } D)=P(D \text { and } H) $$ As you can see from the last equation, conditional probabilities do not commute: $$ P(H \mid D) \neq P(D \mid H) $$