Curious Minds

Recently we were listening to "Hotel California", the Eagles classic, featuring one of the greatest guitarists of all time, Don Felder. The song is great, but lie the gees we are, something in the lyrics caught our ear: "Welcome to the Hotel California   Such a lovely place (such a lovely place)   Such a lovely face.   Plenty of room at the Hotel California   Any time of year (any time of year) you can find it here" The line in bold triggered something in our memory, something we read about when we started learning the concepts of sets and different types of infinities: The Hilbert Infinite Hotel Paradox. We wonder whether the songwriters deliberately wrote this. Anyway, this led us to revisit this paradoxical problem.  The Problem: Consider a hypothetical hotel with countably infinite number of rooms. A lone traveler seeing shelter arrives at the hotel, only to find all rooms occupied. Yet, he is given a room to stay...

The Jeffreys-Lindley paradox is an apparently puzzling problem in Statistical Inference . It has been seen often that frequentist and Bayesian approaches to the testing of a point null hypothesis i.e. a simple hypothesis , lead to divergent results especially when the sample size is large and for different choices of the prior distribution of the parameter under study. Statement of the paradox: The paradox can be understood in the general setting as follows: Let x denote the observation or the data obtained from the experiment under study. A test of significance for the null hypothesis   gets rejected at level of significance  . The posterior probability of  , given the data x , is very high even for small prior probability of  . Lindley's original formulation of the problem in his paper "A Statistical Paradox" published in 1957 may be stated as follows: Suppose we compare different sets of observations with varying sample sizes 'n' , al...