Posterior summaries I classical statistics we have estimators for arameters. These are fuctios of data e.g. mea of observatios or samle variace. Parameter is thought fixed but ukow. Data is radom therefore estimator is radom.
Posterior summaries I Bayes: osterior desity describes our ucertaity about the ukow arameter after observig data. Observed data is fixed it s what it is. evidece. Parameter is radom because it is ucertai. Probability is a measure of ucertaity. Posterior desity is comlete descritio. Mode the most robable value. Mea exected value if you d make a bet. Media with 50% robability it s below this. 2
Posterior summaries Comariso of mea media mode: Defie a loss fuctio Lδ x to describe the loss due to estimatig by oit estimate δ x based o data x. For ay x choose δ x to miimize the osterior loss E L δ x L δ x d x x If the loss fuctio is quadratic Lδ x δ x 2 the the osterior loss becomes V+E-δ x 2 which is miimized by choosig δ x E the osterior mea. 3
Posterior summaries But if our loss fuctio is Lδ x δ x the we should choose δ x osterior media to miimize osterior loss for ay x. Ad if Lδ x {δx } δ x all-or-othig error the the choice would be osterior mode. E.g. if you refer choosig osterior mea this meas that you behave as if you had a quadratic loss fuctio. No oit value ca fully covey the comlete iformatio cotaied i a osterior distributio. 4
Posterior summaries Comare: classical 95% Cof. Iterval? I classical statistics: cofidece iterval is a fuctio of data therefore radom. With 95% frequecy the iterval will cover the true arameter value i the log ru. If the exerimet is reeated. i.e. we are 95% cofidet of this. 5
Posterior summaries 95% Credible iterval. I Bayes: credible iterval is a iterval i which the arameter is with 95% robability give this actual data we ow had. 95 % Ca choose 95% iterval i may ways though. 6
Posterior summaries 95% Credible iterval. Posterior desity ca be bimodal or multimodal. CI does ot eed to be a coected set. 95 % A shortest ossible iterval with a give robability is Highest Posterior Desity Iterval 7
Geerally: Posterior summaries With little data osterior is dictated by rior With eough data osterior is dictated by data Savage: Whe they have little data scietists disagree ad are subjectivists; whe they have iles of data they agree ad become objectivists. V E V + V E which meas that osterior variace V is exected to be smaller tha the rior variace V. But sometimes it ca icrease. 8
Further use of osteriors Hyotheses: About a arameter: <0 Comute P < 0 the cumulative desity at 0. PH 0 ad PH ossible to comute if H is a regio of arameter sace. We do ot reject or accet a H just calculate its robability give evidece. 9
Further use of osteriors Hyotheses: Sometimes used: osterior odds PH 0 /PH. If > shows suort for H 0. Bayes factor: a ratio of rior ad osterior odds BF [ PH 0 /PH ] / [ PH 0 /PH ] [PH 0 PH ] / [PH PH 0 ] Posterior odds Prior odds x BF This is a differet way of exressig Bayes theorem: BF exresses how much data chage rior odds. 0
Further use of osteriors Hyotheses: A oit hyothesis H 0 : 0 agaist H : We must have ositive robability PH 0 -PH The BF the becomes the same as likelihood ratio Because costat cacels out. But: how big small BF is big small eough? Comosite hyothesis oe-sided two-sided 0 0 0 P P P P
Further use of osteriors ~ N data:.5 A oit hyothesis H 0 : 0 agaist H : 2. Assume rior 020.5 The osterior odds likelihood ratio. Coversio to robability : /+odds 0 0 0 P P P P 2
Further use of osteriors Predictios: It is rather easy to comute redictive distributio of based o give arameters ad the model P. Ad likewise for ay fuctio g. - Assumig you ca geerate samles from P. This would ot take ito accout the ucertaity about arameters. Aim: to comute osterior redictive distributio P ew obs This gives redictio based o the ast data ot based o assumed arameter estimates. 3
Predictive distributios Cosider series of observatios: ad a model i so that i are coditioally ideedet give. Posterior redictive distributio of + : + + + + d d d Our model Posterior of 4
Predictive distributios Likewise: Prior redictive distributio of + : + d d + + Our model Prior of With the redictive aroach arameters dimiish i imortace esecially those that have o hysical meaig. From the Bayesia viewoit such arameters ca be regarded as just lace holders for a articular kid of ucertaity o your way to makig good redictios. Draer 997 Lidley 972. 5
Predictive distributios Note also directly from Bayes: by isertig rior osterior model of we fid rior redictive desity of. Similarly + + 6
Predictive distributios Let s try with biomial model. Assume we have a osterior which is beta. Old data is the icluded i. This ca be solved as: A + B N-+ A BETA-BINOMIAL distributio. d d BiomialN Beta B A B A N + Γ Γ Γ Γ Γ + Γ 7
8
Predictive distributios With Poisso model: Assume we have a rior which is gamma. If osterior old data is icluded i. The solutio is NEGATIVE BINOMIAL distributio: λ λ λ λ d d Poissoλ gamma + + + 9
Predictive distributios To solve redictive meas variaces: Use E EE Use V EV + VE For examle with Poisso + Gamma: E / V / + / 2 By icludig arameter ucertaity to a model we get models d suitable for e.g. overdisersed data. 20