122S:138Bayesian StatisticsLecture 8Sept. 18, 2009Kate Cowles374 SH, [email protected] predictive distribution of afuture observation• p(ynew|y) is Normal– mean is posterior mean– variance is sum of∗ data variance σ2(assumed known in thisunrealistic case)∗ variance of posterior mean3Jeffreys prior for normal mean with datavariance assumed known• p(µ) ∝ 1, −∞ < µ < ∞• limit of N(µ0, σ20) as σ20goes to ∞σ20is prior variance• equivalently, limit of N(µ0, τ20) as τ20goes to0τ20is prior precision4What is the posterior predictive dis-tribution of the healing rate of a newnewt?5Inference about the spread of a normaldistribution• primary research question may concern vari-ability of response varia ble in population– quality control in industry– reponse to medical treatment6Inference for the variance of a normaldistribution• Suppose in the newt he a ling rate examplethat we knew the population mean µ = 25but we did not know the population va ri anceσ2• Equivalently, we don’ t know the populationprecision τ2• We wish to infer about th e distribution s ofthese parameters that describe the spread ofthe normal distribution.7• Recall the joint sampl ing distribution of nobservations modelled as conditionally i nde-pendent draws from a normalp(y1, . . . , yn|µ, σ2) =nYi=11√2πσexp−(yi− µ)22σ2∝1(σ2)n2exp−Pni=1(yi− µ)22σ2• We will assume (unrealistically) that µ is aknown constant8Sufficient statistic• sufficient statistic for σ2isP(yi− µ)2• we can write likelihood equivalently asp(y|σ2) ∝1(σ2)n2exp−nv2σ2, 0 < σ2< ∞wherev =1nX(yi− µ)2• What is corresponding co njugate prior?9Inverse gamma distributionWhat is p osterior?p(σ2|y) ∝ ?10Alternative parameterization of priorp(σ2) = IG(ν02,ν0σ202)Then we can think of prior as providing equiv-alent information to• ν0prior observations• with σ20average squared deviation from knownµ11Estimating σ2of healing rates in pop-ulation of newts• Suppose µ was known to be 25.• Suppose we had previously studi ed 2 newtsand the average squa red difference betweentheir healing rate and 2 5 was 64.• What is our appropriate prior?• for newt dataP18i=1(yi− 25)2= 1201• What is posteriorp(σ2|y) ∝• Recognize this as?12Noninformative prior for normal vari-ance• What inverse gamma prior would have info r-mation equivalent to 0 prior observations?• How would you write this as a p.d.f for σ2?• Is it proper or improper?• What characteristics would a dataset have tohave in order to produce a proper posterio rdistribution for σ2if this prior were used?13Priors for normal precision• ifσ2∼ IG(α, β)andτ2=1σ2,thenτ2∼ G(α, β)• You must be careful of parameterizations ofboth gamma and inverse gamma
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