week 71 Hypothesis Testing Suppose that we want to assess the evidence in the observed data, concerning the hypothesis. There are two approaches to assessing this hypothesis. We could compute the posterior probability and if this is small, we conclude that we have evidence against H 0. There is one major problem with the approach above. For when the prior distribution of ψ is absolutely continuous, then we have that for all data s. Therefore we would always find evidence against H 0 no matter what data we observe. To avoid this problem, there is an alternative approach to hypothesis assessment that is sometime used.

week 72 Example Indicator function example done in class…

week 73 Bayesian P-value Recall that, if ψ 0 is a surprising value for the posterior distribution of ψ then this is evidence that H 0 is false. The value ψ 0 is a surprising whenever it occurs in a region of low probability for the posterior distribution of ψ. A region of low probability will correspond to a region where the posterior density is relatively low. So one possible method for assessing this is by computing the Bayesian P-value given by Note that when the posterior density of ψ is unimodal, then the Bayesian P-value corresponds to computing a tail probability.

week 74 Interpretation If the above P-value is small, then ψ 0 is a surprising, at least with respect t0 our posterior beliefs. If we decide to reject H 0 whenever the P-value is less than 1-α, then this approach is equivalent to computing a α-HPD region for ψ and rejecting H 0 whenever ψ 0 is not in the region.

week 75 Example Suppose that the posterior distribution of θ is Beta(2,1). The density of θ is then… Further suppose that we want to assess H 0 : θ = ¾. Then…

week 76 Odds Ratio In a probability model with sample space S and probability measure P, the odds in favor of event is defined to be This is the ratio of the probability of A to the probability of A c. Large values of the odds in favor of A indicate that there is a stronger belief that A is true.

week 77 Bayes Factors Bayes factors are another method of hypothesis assessment. The Bayes factors in favor of the hypothesis H 0 is defined to be the ratio of the posterior odds in favor of H 0 to the prior odds in favor of H 0 or Note, this is defined whenever the prior probability of H 0 is not 0 or 1.

week 78 Interpretation of Bayes Factor Bayes factor in favor of H 0 is measuring the degree to which the data changed the odds in favor of the hypothesis. If is small, then the data are providing evidence against H 0 and evidence in favor of H 0 when is large. It is not immediately clear how to interpret the actual value of in particular how large does it has to be to provide strong evidence in favor of H 0. One approach to this problem is to use the relationship between the posterior probability of H 0 being true and. It is given in the form is the prior odds in favor of H 0. So when Bayes factor is small, then the posterior probability of H 0 is small and conversely.