Inference for deterministic simulation models: The Bayesian melding approach

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SeptemberSpeed TModelling CSIRO DMS NewsletterTerrell G RThe maximal smoothing principle in density estimation J Amer StatistAssWade P R and DeMaster D PA Bayesian analysis of eastern Paci c gray whale populationdynamics Paper SC AS presented to the IWC Scienti c Committee JuneWolpert R LComment J Amer Statist Ass AppendixProof of Theorem The joint distribution ofispp ggdgddgdand rewriting the model M in terms of and with as the output yieldsg M gNow consider the transformationU V whereUg M gVso thatVU g M g Vwith associated Jacobian g M g vvIt follows that the joint distribution of U and V ispU V u vp v g M g vp gv g u g M g vdg vdvdgdu g M g vNow noting that Ug M g vM we havepV jU v jpU V vp gv M g vdg vdvdgdg M g vSince Vand conditioning on Uis equivalent to conditioning on the modelthis isexactly the Bayesian synthesis post model distribution of Writing in terms of we havep gM gdgddgdg M gTo express the post model distribution in terms of we transformwhich has inverseJacobian dgd Noting the identitiesdgddgdanddgddgdwe obtain p MdgdMas in Proof of Theorem We rst consider the continuous case Forandthe resultis trivially true from the assumptions Forthe function h x x is concave onbecause h xxNoting that the ratio qq has rangeforn f qqg and applying Jensen s inequality we haveZqqdEqqqEqqq sinceEqqqZqdSoRqqdand the normalizing constant k can be found Note thatthe proof proceeds analogously in the discrete case except that the integrals are replaced by theappropriate sums Proof of Theorem We rst note thatSAiSCj and that the Cj are mutually exclusive becauseeach Ai appears in only one Cj namely Cm i Since!q Bm iand q Ai q Bm iit follows that !q Aifor each i By the mutual exclusivity of the CjXi!q AiXjXi Ai Cj!q AiXjXi Ai Cj!q Bjq AiQ CjXjXi Ai Cj!q Bjq AiPk Ak Cj q AkXj!q Bj