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4 months ago in Probability Theory By Meera

Does bayes’ theorem still work if the sample space keeps changing?

I’m analyzing a clinical trial with adaptive enrollment and I’m updating posterior probabilities as data arrives. But unlike a fixed urn problem, my “sample space” here the set of patients and their possible outcomes is literally expanding and contracting. I can compute P(A|B) mechanically, but I’m uneasy. Am I violating a core assumption of Bayes if the space isn’t static?

SampleSpace

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By Hema Answered 1 month ago

It breaks down. Bayes' theorem assumes a fixed, common sample space for both events. It's a clean swap of conditionals—P(A|B) from P(B|A) but that swap only works if A and B live in the same probabilistic universe. If the sample space itself is dynamically changing between observations, the foundational relationship dissolves. You're no longer doing Bayes; you're doing something else entirely.

By Raman Answered 2 months ago

From my experience in biostatistics, this exact worry comes up constantly in adaptive trial design. Mathematically, Bayes’ theorem is just a relationship between conditional and marginal probabilities defined within a single, fixed probability space. If your sample space literally changes new patients added mid?analysis you are no longer conditioning on events from the original space. What saves you is re?defining your probability space at each update to include all potential participants who could have been observed up to that time. I recommend framing your problem hierarchically: model the enrollment process explicitly. If enrollment is ignorable (non?informative), sequential updating works fine. If it’s outcome?dependent, you need a joint model for both the data and the sampling mechanism itself.

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Does bayes’ theorem still work if the sample space keeps changing?

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