Lindley's Paradox

Glenn Shafer
1982 Journal of the American Statistical Association  
A sharp null hypothesis may be strongly rejected by a standard sampling-theory test of significance and yet be awarded high odds by a Bayesian analysis based on a small prior probability for the null hypothesis and a diffuse distribution of one's remaining probability over the alternative hypothesis. This disagreement between sampling-theory and Bayesian methods was first studied by Harold Jeffreys (1939), and it was first called a paradox by Dennis Lindley (1957). The paradox can be exhibited
more » ... x can be exhibited in the simple case where we are testing θ = 0 using a single observation Y from a normal distribution with variance one and mean θ. If we observe a large value y for Y (y = 3, for example), then standard sampling theory allows us to confidently reject the null hypothesis. But the Bayesian approach advocated by Jeffreys can give quite a different result. Jeffreys advised that we assign a non-zero prior probability π 0 to the null hypothesis and distribute the rest of our probability over the real line according to a fairly flat probability density π 1 (θ). If the range of possible values for θ is very wide, then the set of values within a few units of y will be very unlikely under π 1 (θ), and consequently the overall likelihood of the alternative hypothesis, exp(-(y-θ) 2 /2)π 1 (θ)dθ ,
doi:10.1080/01621459.1982.10477809 fatcat:xdjxnhd7jjb7jcpo4to7nxhgjq