figure the bic and marginal likelihood used for model selection it is possible to give the model complexity parameter bayesian treatmentin which we define prior density on the set of all models under consideration for examplelet ( ) be prior density on candidate models treating the model complexity index as an additional parameter to th and applying bayesformulathe posterior for (thpcan be written asg(thp tg(th ptx ( tg( pg(pg( thpg(th px ( pg( { { posterior of th given model posterior of model the model evidence for fixed is now interpreted as the prior predictive density of tconditional on the model pz ( pg( thpg(th pdthp and the quantity (tmp= ( pg(pis interpreted as the marginal likelihood of all the candidate models finallya simple method for model selection is to pick the index with the largest posterior probabilityb argmax ( targmax ( pg(pp example (polynomial regression (cont )let us revisit example by giving the parameter mwith bayesian treatment recall that we used the notation in that example we assume that the prior ( / is flat and uninformative so that the posterior is given by ( yg( | | / ( / | | / ( ( ) /