Discussion Overview
The discussion revolves around the application of conditional probability, specifically focusing on the expression Pr(Z|X&Y) and its relationship to Bayes' Rule. Participants explore the implications of independence between events X and Y in the context of conditional probabilities.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant proposes that Pr(Z|X&Y) can be expressed using Bayes' Rule as Pr(Z|X&Y)=[Pr(X&Y|Z)Pr(Z)]/Pr(X&Y).
- Another participant challenges the assumption that independence of X and Y implies Pr(X&Y|Z)=Pr(X|Z)Pr(Y|Z), providing a counterexample involving coin tosses where Z creates dependence between A and B.
- A later reply acknowledges a mistake in dividing by the conditional and questions whether Bayes' Rule still applies in the context of joint events X and Y.
- One participant clarifies that Bayes' Rule is valid as long as P(A) is not zero, noting potential issues with division by zero in certain cases.
Areas of Agreement / Disagreement
Participants express differing views on the implications of independence given a condition Z, with some asserting that independence does not hold under conditioning, while others maintain that Bayes' Rule is applicable in the presented scenario. The discussion remains unresolved regarding the specific application of independence in this context.
Contextual Notes
There are limitations regarding the assumptions made about independence and the conditions under which Bayes' Rule is applied. The discussion does not resolve the mathematical steps involved in the application of these concepts.