(sadly I seem to be unable to propose edits directly) The proof involved deriving three equalities:
1. P(Hj)/P(Hk) * P(e0|Hj)/P(e0|Hk) = P(e0∧HJ)/P(e0∧Hk) so far so good. This derives directly from the definition of conditional probability, P(Y)*P(X|Y)=P(X∧Y), when applied to both the numerator and denominator of the fraction on the left hand side.
2. P(e0∧Hj)/P(e0∧Hk) = (P(Hj|e0)/P(e0)) / (P(Hk|e0)/P(e0)) This looks like a roundabout application of the definition of conditional probability. While it technically holds, it seems to me like it follows less directly from P(X∧Y)=P(Y)*P(X|Y) than the following would: P(e0∧Hj)/P(e0∧Hk) = (P(Hj|e0)*P(e0)) / (P(Hk|e0) * P(e0))
3. the final equality would then change from (P(Hj|e0)/P(e0)) / (P(Hk|e0)/P(e0)) = P(Hj|e0) / P(Hk|e0) to (P(Hj|e0)*P(e0)) / (P(Hk|e0) * P(e0)) = P(Hj|e0) / P(Hk|e0)
What a nice explainer this has been so far! I do think the proof given for Bayes’ rule in odds form has room for improvement. Not sure if it shows up in every ‘path’, so I’ll link the specific url I visited here: https://www.lesswrong.com/w/bayes-rule-odds-form?pathId=61b&lens=introduction-to-bayes-rule-odds-form
(sadly I seem to be unable to propose edits directly)
The proof involved deriving three equalities:
1.
P(Hj)/P(Hk) * P(e0|Hj)/P(e0|Hk) = P(e0∧HJ)/P(e0∧Hk)
so far so good. This derives directly from the definition of conditional probability, P(Y)*P(X|Y)=P(X∧Y), when applied to both the numerator and denominator of the fraction on the left hand side.
2.
P(e0∧Hj)/P(e0∧Hk) = (P(Hj|e0)/P(e0)) / (P(Hk|e0)/P(e0))
This looks like a roundabout application of the definition of conditional probability. While it technically holds, it seems to me like it follows less directly from P(X∧Y)=P(Y)*P(X|Y) than the following would:
P(e0∧Hj)/P(e0∧Hk) = (P(Hj|e0)*P(e0)) / (P(Hk|e0) * P(e0))
3.
the final equality would then change from
(P(Hj|e0)/P(e0)) / (P(Hk|e0)/P(e0)) = P(Hj|e0) / P(Hk|e0)
to
(P(Hj|e0)*P(e0)) / (P(Hk|e0) * P(e0)) = P(Hj|e0) / P(Hk|e0)
Putting it all together into a single line:
P(Hj)/P(Hk) * P(e0|Hj)/P(e0|Hk) = P(e0∧HJ)/P(e0∧Hk) = (P(Hj|e0)*P(e0)) / (P(Hk|e0) * P(e0)) = P(Hj|e0) / P(Hk|e0)