for model is the same as that for model conditioned on that iswe assume ( ( as model contains model as special casethe latter is said to be nested inside model we can formally write (see also exercise ) ( ( up ( du ( un ( du ( hencethe bayes factor simplifies to savage-dickey density ratio (tn ( ( (tn ( tp ( ( ( ( in other wordsb is the ratio of the posterior density to the prior density of nevaluated at and both under the unrestricted model this ratio of posterior to prior densities is called the savage-dickey density ratio whether to use classical (frequentistor bayesian model is largely question of convenience classical inference is useful because it comes with huge repository of readyto-use resultsand requires no (subjectiveprior information on the parameters bayesian models are useful because the whole theory is based on the elegant bayesformulaand uncertainty in the inference ( confidence intervalscan be quantified much more naturally ( credible intervalsa usual practice is to "bayesifya classical modelsimply by adding some prior information on the parameters further reading popular textbook on statistical learning is [ accessible treatments of mathematical statistics can be foundfor examplein [ ][ ]and [ more advanced treatments are given in [ ][ ]and [ good overview of modern-day statistical inference is given in [ classical references on pattern classification and machine learning are [ and [ for advanced learning theory including information theory and rademacher complexitywe refer to [ and [ an applied reference for bayesian inference is [ for survey of numerical techniques relevant to computational statisticssee [ exercises suppose that the loss function is the piecewise linear function loss( , ya ( ) ( )ab where cis equal to if and zero otherwise show that the minimizer of the risk `(ge loss(yg( )satisfies [ (xx xa+ in other wordsg(xis the /( bquantile of yconditional on