What values of the generalized likelihood ratio A are necessary to reject the null hypothesis at the significance level Â = .1 if the degrees of freedom are 1, 5, 10, and 20?
This problem introduces a Bayesian perspective on the likelihood ratio, which will be expanded upon in chapter 15. According to the Bayesian model, one's belief in a hypothesis is represented as a probability; in this manner let P(H1) and P(H2) represent an experimenter's belief in two incompatible hypotheses prior to gathering data, X. They are called prior probabilities. Having observed X, the probabilities change to P(H1IX) and P(H2IX), which are called posterior probabilities. Show that
that is, that to find the ratio of posterior probabilities; the ratio of prior probabilities is multiplied by the likelihood ratio. In the case that H2 is the hypothesis that H1is false, the ratios are the prior and posterior odds of H1.