Bayesian testing and model criticism

we have said that we cannot state with confidence that 0 does not take any particular value that lies with a HPD credible region. While this argnment is common, it falls outside 

We begin by considering, after taking the data Y, which of two hypotheses is more probably tme.hetibenull hypothesis Hobetha.t 0 e Qo C andlctthe alternative hypothesis H be that 0 e c Often, S2i is the complement set to f2o so that H is the hypothesis that 0 S2o. The prior and posterior probabilities of each hypothesis being trne are 

If 5oi (T) 1, Ho is favored by the data. If Bo (T) -C 1, 7/i is favored. The exact value of Bo (Y) is quite sensitive to the choice of prior, and in particular, we cannot use improper priors here. Let us say that we have used the simple fix (8.71) of setting p(0) to zero outside of 2 and to a uniform value inside S2e. Then, P Hj) is merely the ratio of the volume of 2 to that of S2g, and is tiius highly sensitive to the (subjective) choice of 2. For an improper prior with of finite volume, P Hj) is zero and the Bayes factor is not defined. 

It is common in frequentist statistics to consider a point-null hypothesis that 0 takes exactly a specific value 0. When 0 is continuous, such a hypothesis has zero probability of being true and is ill posed in the Bayesian framework. We can approximate such a point-null hypothesis as Hq [e o-0 II s, as long as the prior is proper. 

Similar techniques are used to judge which of several models best fits the data. Let us propose several models a = 1, 2. of which one is believed to be the true model. Let Ma be the event that model a is the true one, and let 7 be the event that we observe the particular data set Y. Then, Bayes theorem applied to the joint probability P(Ma n Y) yields the posterior probability that model a is the true one, given the data Y  

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