About the Wrong Proof of the Lindley Paradox

Fausto Galetto (1)

1 Politecnico di Torino DIGEP

The purpose of this paper is to show that the so called Lindley Paradox is not at all a paradox. It came from a wrong application of Bayesian Theory to the Hypothesis Testing. Two hypotheses H 0 and H 1 are considered, H 0 : {= 0 } versus H 1 : { 0 }; denotes the parameter whose value we want to test. Two theories are available: Classical Statistics and Bayesian Statistics. Lindley (in 1957) pretended to prove that the two theories provide two opposite and contradictory solutions to the same problem: that, obviously, is not scientifically acceptable. Using the "Distribution Theory (in Mathematics)" which extends the Theory of the Real Functions to the Real Functionals, where the "Dirac Delta " (named also the "Impulse Function") is scientifically defined and it is the derivative of the Heaviside function, we find the true form of the posterior probability P(H 0 | data) and prove the mistake of Lindley. Researchers must be alert in order to do a good job…. Everybody have to consider the methods of Logic and of the Scientific Theory (Mathematics, Probability, Statistics, Physics…). Several Professors do not practice them.

Mots clés: Bayesian approach; Quality Methods and Statistics; Scientific Approach; Intellectual Honesty

Domaines: Théorie [stat.TH]; Mathématiques [math]

Fausto Galetto. About the Wrong Proof of the Lindley Paradox. 2021. ⟨hal-03479046⟩