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I don't remember it exactly, and I'd like to read up on it again.

It is about experiment design and frequentist hypothesis testing. A sequence of N measurements are made, and then when evaluating the data, whether the data shows a significant effect depends on why the experiment was a terminated: because it was already predecided that N measurements need to be made, or because of some other criterion (I don't remember). The "paradox" is that the data is the same, but we have to know how the experiment would have continued if something else had happened. So a part of the plan that never got applied is required to decide whether the effect is significant.

I clearly remember that this has a detailed Wikipedia article, but I can't find it either through Google or Wikipedia search or browsing. Maybe it was not designated as a "paradox", but an effect, phenomenon, thought experiment or something else.

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You might be thinking of the Likelihood Principle.

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As @Flounderer mentioned, this is related to the likelihood principle. I think that, even more specifically, the "paradox" you're referring to is a violation of the conditionality principle.

The conditionality principle is attractive, since it does seem strange that the interpretation of experimental results would depend on counterfactual results from experiments that were not actually run, but the principle is not universally accepted -- notably, it is not accepted by frequentists.

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You may be referring to the literature surrounding sequential hypothesis testing. This field originated with Abraham Wald during WWII and publication of his book Sequential Analysis around 1948 ... https://www.amazon.com/Sequential-Analysis-Abraham-Wald/dp/0486615790/ref=sr_1_1?s=books&ie=UTF8&qid=1472844126&sr=1-1&keywords=abraham+wald+sequential

More recently, Alexander Tartakovsky's book, also titled Sequential Analysis has become something of the bible in this field for its comprehensive overview ... https://www.amazon.com/Sequential-Analysis-Hypothesis-Changepoint-Probability-ebook/dp/B00MMOIWTS/ref=sr_1_1?s=books&ie=UTF8&qid=1472844136&sr=1-1&keywords=tartakovsky+sequential

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