Bản quyền thuộc Đại học Cambridge năm 2003. Xem trên màn hình cho phép. In ấn không được phép. Bạn có thể mua cuốn sách này cho £ 30 hoặc $ 50. Xem cho các liên kết. | Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http 0521642981 You can buy this book for 30 pounds or 50. See http mackay itila for links. The bent coin and model comparison 53 Model comparison as inference In order to perform model comparison we write down Bayes theorem again but this time with a different argument on the left-hand side. We wish to know how probable H1 is given the data. By Bayes theorem P HI s F P s 1 F H1 P H1 H1 Is F P s IF 3-1Z Similarly the posterior probability of Ho is PWn Is P s 1 F Ho P Ho z 1 P Ho 1 s F ----P s F ----- 3-18 The normalizing constant in both cases is P s I F which is the total probability of getting the observed data. If H1 and Ho are the only models under consideration this probability is given by the sum rule P s I F P s I F H1 P H1 P s I F Ho P Ho To evaluate the posterior probabilities of the hypotheses we need to assign values to the prior probabilities P H1 and P Ho in this case we might set these to 1 2 each. And we need to evaluate the data-dependent terms P s I F H1 and P s I F Ho . We can give names to these quantities. The quantity P s I F H1 is a measure of how much the data favour H1 and we call it the evidence for model H1. We already encountered this quantity in equation where it appeared as the normalizing constant of the first inference we made - the inference of pa given the data. How model comparison works The evidence for a model is usually the normalizing constant of an earlier Bayesian inference. We evaluated the normalizing constant for model H1 in . The evidence for model Ho is very simple because this model has no parameters to infer. Defining Po to be 1 6 we have P s I F Ho pF 1 - Po Fb Thus the posterior probability ratio of model H 1 to model Ho is P H I s F P Ho I s F P s I F H1 P H1 P s I F Ho P Ho Fa Fb Fa Fb 1 P Some values of this posterior probability ratio
