conditioning on zero probability events

An interesting question on X validated as to how come a statistic T(X) can be sufficient when its support depends on the parameter θ behind the distribution of X. The reasoning there being that the distribution of X given T(X)=t does depend on θ since it is not defined for some values of θ … Which is not correct in that the conditional distribution of X depends on the realisation of T, meaning that if this realisation is impossible, then the conditional is arbitrary and of no relevance. Which also led me to tangentially notice and bemoan that most (Stack) exchanges on conditioning on zero probability events are pretty unsatisfactory in that they insist on interpreting P(X=x) [equal to zero] in a literal sense when it is merely a notation in the continuous case. And undefined when X has a discrete support. (Conditional probability is always a sore point for my students!)

2 Responses to “conditioning on zero probability events”

  1. “it is merely a notation” — that seems to be the case since conditioning on “events” with probability zero is commonly seen. Is there a good clarification on what is under the hood?

    • The proper approach to defining conditional distributions is the one based on measure theory, that is with the conditional expectation being a measurable function of the conditioning rv producing the same expectation as the unconditional one.

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