consider the bayesian model for { xn with likelihood ( usuch that ( xn ~iid ( and prior pdf (usuch that ( for some hyperparameter define sequence of densities wt ( ) via wt (uwt- (ug( )starting with (ug(ulet at and bt denote the mean and precision of under the posterior gt ( tg( )wt (ushow that gt ( tis normal density with precision bt bt- nb and mean at ( gt )at- gt xn nwhere gt : /(bt- nhencededuce that gt ( tconverges to degenerate density with point-mass at xn consider again example where we have normal model with improper prior (thg(us / show that the prior predictive pdf is an improper density ( but that the posterior predictive density is ( xn ) ( ( ) !- / -xn deduce that ( + )/( - tn- iid assuming that xn show that ( holds and that ` - ln ( suppose that { xn are observations of iid continuous and strictly positive random variablesand that there are two possible models for their pdf the first model is ( thp th exp (-thxand the second is th ( thp ! / thx exp for both modelsassume that the prior for th is gamma density (thbt - th exp (-bthg(twith the same hyperparameters and find formula for the bayes factorg( )/ ( )for comparing these models suppose that we have total of possible models with prior probabilities ( ) show that the posterior probability of model ( tcan be expressed in terms of all the ( bayes factors- ( jg( ( ij, the precision is the reciprocal of the variance