tag:blogger.com,1999:blog-715339341803133734.post792228013485891804..comments2017-03-18T13:40:54.923-05:00Comments on Maximum Entropy: Whose confidence interval is this?Tom Campbell-Rickettshttp://www.blogger.com/profile/07387943617652130729noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-715339341803133734.post-4372877483382214832014-03-22T10:07:54.455-05:002014-03-22T10:07:54.455-05:00Thanks for elaborating.
Yes, frequentist thinking...Thanks for elaborating.<br /><br />Yes, frequentist thinking is complicated - it needs to be to disguise the fact that it is wrong. The device that the frequentists use to make e.g. item 4 false is completely ad hoc, declared out of thin air, and has no basis. Tom Campbell-Rickettshttp://www.blogger.com/profile/07387943617652130729noreply@blogger.comtag:blogger.com,1999:blog-715339341803133734.post-21045046307686199552014-03-22T05:55:14.721-05:002014-03-22T05:55:14.721-05:00To be really precise you should note that sentence...To be really precise you should note that sentences of the form "[0.1,0.4] is a 95% confidence interval" are never legit. You can only assign the property of being a "95% confidence interval" to <i>procedures</i> that calculate an interval given the data. The best you can say is "[0.1,0.4] was generated by a procedure that (before I knew the data) I assigned a 95% probability of creating an interval that contained the true value".<br /><br />So we have:<br /><br />False: 4a. <i>There is a 95% probability that the true mean lies between 0.1 and 0.4.</i><br /><br />True: 4b. <i>There is a 95% probability that the true mean lies between the lower end of the confidence interval and the higher end (provided we haven't yet seen the data that determines what they are).</i><br /><br />Our problem here is that the frequentists refuse to treat the true value as an r.v. So before we get the data the Bayesian views both the true value and data as random, but the frequentist views only the data as random. So at this point the frequentist can say "performing the following procedure on the data will yield an interval that with 95% probability contains the true value". When they say this it is the <i>interval</i> that they are treating as random, and they are saying that the statement is true <i>for each possible true value</i>. The Bayesian is treating both the true value and interval as random, but actually agrees with the frequentist's statement since if the statement is true for each possible true value then it must also be true for the random true value weighted according to the Baysian's prior.<br /><br />After we find out the data, the frequentist calculates her confidence interval <i>and then has no random variables left at all</i>. Thus the frequentist is now incapable of making any probabilistic statements at all. And so all six of the given statements are false.<br /><br />Now, what does the Bayesian do when we find the data? She updates her probability distribution for the true value. Also, just like for the frequentist, the interval can be calculated and so ceases to be an r.v. But the Bayesian can still make some probabilistic statements since for her the true value is still random. In particular the Bayesian can calculate the probability, based on their posterior, that the true value lies in the interval. But this needn't be 95%, and so even from a Bayesian perspective we must judge all six propositions to be false.<br /><br />An amusing example is to consider an experiment where the true value is known to be positive (perhaps it is a scale parameter) but it is being measured with some Gaussian noise (with say s.d. 1). Then it is clear that a 95% confidence interval will be given by taking the measured value plus or minus 1.96. So suppose by misfortune we get the measurement "-2" then our confidence interval is [-3.96,-0.04]. Certainly no one would claim that our true value was 95% certain to lie in there!<br /><br />Gosh, frquentist thinking is complicated isn't it?Oscar Cunninghamnoreply@blogger.com