Talk:Lindley's paradox
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[edit] question
Can someone give some concrete examples demonstrating this paradox. I find the article a bit difficult to understand. —The preceding unsigned comment was added by 24.227.161.246 (talk • contribs) 2007-05-26T02:37:18.
- There are two different ways of interpreting the results of a statistical experiment, the so-called "frequentist" and "Bayesian" approaches. These two ordinarily produce similar results in practice. In some settings, however, the two approaches produce completely opposite results, to the point where one of them would give a strong answer in favor of the hypothesis while the other would give a strong answer against it. CMummert · talk 12:12, 26 May 2007 (UTC)
[edit] Better example?
I feel like the gravity/airplane example does not really demonstrte Lindley's paradox. It is stated that the prior weakly supports the null hypothesis, and that the evidence causes opposite Bayesian and frequentist conclusions. Neither of these occurs in the gravity example, where the prior on H0 is strong, and where the evidence reduces our belief in both paradigms (just to different extents -- in the example, the frequentist rejects it, while Bayesian just barely reduces his belief). But that seems to miss the underlying paradox. Could we have a better example?
Maybe someone could flesh out an example like the following. Suppose theoretical physicists have two competing "theories" of unification - call them string theory and the loop model (I'm making this up - don't take this to be an accurate characterization of actual physics). Suppose we take the null hypothesis that the loop model is correct to be p(H0)=0.001. (i.e., The odds of it being correct is small). String theory, however, is actually a class of theories, rather than a single theory, and in fact, let's assume that it is an infinite class of theories (i.e., there are some parameters, and for a particular choice of parameters, you end up one one possible instantiation of a string theory). So we have a probability density over all these potential string theories, with no single instantiation having a strictly positive probability.
Now, we make some observation. It could be any observation with a non-zero likelihood under each theory, e.g., p(x|H)>0 for H=loop and for H=any string instantiation, but also with p(x|H0)<5%. The frequentist rejects H0 -- concludes that loop theory is false. Now, as the entropy of the prior (over the string theories) increases, the posterior p(H0|x) gets arbitrary close to 1. So the Bayesian becomes more inclined to accept H0 (loop theory). I think this is the essence of the paradox -- the observation (and it doesn't really matter what the observation is) causes the Bayesian to become very confident the null hypothesis is true (even with a very small prior), while the frequentist sees the observation as a reason to reject it. Someone needs to flesh out this, or some other example, a little more cleanly and make it understandable to the wiki audience before it would be suitable for the wiki page, but I propose that an example along those lines would demonstrate the essence of the paradox better than the current example.
