Resolving Jeffreys-Lindley Paradox

Abstract

Jeffreys-Lindley paradox is a case where frequentist and Bayesian hypothesis testing methodologies contradict with each other. This has caused confusion among data analysts for selecting a methodology for their statistical inference tasks. Though the paradox goes back to mid 1930's so far there hasn't been a satisfactory resolution given for it. In this paper we show that it arises mainly due to the simple fact that, in the frequentist approach, the difference between the hypothesized parameter value and the observed estimate of the parameter is assessed in terms of the standard error of the estimate, no matter what the actual numerical difference is and how small the standard error is, whereas in the Bayesian methodology it has no effect due to the definition of the Bayes factor in the context, even though such an assessment is present. In fact, the paradox is an instance of conflict between statistical and practical significance and a result of using a sharp null hypothesis to approximate an acceptable small range of values for the parameter. Occurrence of type-I error that is allowed in frequentist methodology plays important role in the paradox. Therefore, the paradox is not a conflict between two inference methodologies but an instance of not agreeing their conclusions