Bayes AI

Unit 5: Bayesian Hypothesis Tests

Vadim Sokolov
George Mason University
Spring 2025

Lindley’s Paradox

Often evidence which, for a Bayesian statistician, strikingly supports the null leads to rejection by standard classical procedures.

  • Do Bayes and Classical always agree?

Bayes computes the probability of the null being true given the data \(p ( H_0 | D )\). That’s not the p-value. Why?

  • Surely they agree asymptotically?

  • How do we model the prior and compute likelihood ratios \(L ( H_0 | D )\) in the Bayesianwork?

Bayes \(t\)-ratio

Edwards, Lindman and Savage (1963)

Simple approximation for the likelihood ratio. \[ L ( p_0 ) \approx \sqrt{2 \pi} \sqrt{n} \exp \left ( - \frac{1}{2} t^2 \right ) \]

  • Key: Bayes test will have the factor \(\sqrt{n}\)

This will asymptotically favour the null.

  • There is only a big problem when \(2 < t < 4\) – but this is typically the most interesting case!

Coin Tossing

Intuition: Imagine a coin tossing experiment and you want to determine whether the coin is “fair” \(H_0 : p = \frac{1}{2}\).

There are four experiments.

Expt 1 2 3 4
n 50 100 400 10, 000
r 32 60 220 50 98
\(L(p_0)\) 0.81 1.09 2.17 11.68

Coin Tossing

Implications:

  • Classical: In each case the \(t\)-ratio is approx \(2\). They we just \(H_0\) ( a fair coin) at the 5% level in each experiment.

  • Bayes: \(L ( p_0 )\) grows to infinity and so they is overwhelming evidence for \(H _ 0\). Connelly shows that the Monday effect disappears when you compute the Bayes version.