By Insua D.R., Ruggeri F., Wiper M.P.

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New Delhi: New Age International. M. (2000) Stochastic Processes: Inference Theory. Dordrecht: Kluwer. E. I. (2005) Gaussian Processes for Machine Learning. Cambridge, MA: The MIT Press. G. and Williams, D. (2000a) Diffusions, Markov Processes and Martingales: Volume 1 Foundations. Cambridge: Cambridge University Press. G. and Williams, D. (2000b) Diffusions, Markov Processes and Martingales: Volume 2 Ito Calculus. Cambridge: Cambridge University Press. Ross, S. (1995) Stochastic Processes. New York: John Wiley & Sons, Inc.

N , with m (i) 1 = 1/N , n = 1. 2. While n ≤ M : Approximate f (θ n |Dn ), up to a normalizing constant K, with the mixture N f (θ n |θ (i) n−1 ) f (x n |θ n ). K i=1 (i) (i) To do so, generate θ ¯(i) n ∼ f n−1 (θ n−1 , Wn−1 ), i = 1, . . , N with ¯(i) importance weights m in = f (xn |θ ¯(i) n )( j p(x n |θ n )) Sample N times independently with replacement from (i) (i) {θ ¯(i) n , m n } to produce the random measure {θ n , 1/N }. n = n + 1. Carpenter et al. (1999), Pitt and Shepard (1999), and Del Moral et al.

Then, we could aim at approximating the quantity of interest through m ˜ P(A) = P(A|θ i ) pi , i=1 P1: TIX/XYZ JWST172-c02 P2: ABC JWST172-Ruggeri 34 March 2, 2012 8:45 BAYESIAN ANALYSIS OF STOCHASTIC PROCESS MODELS satisfactorily under appropriate conditions. We first determine the order m ≥ 1 of this reduced order model (ROM), based on purely computational reasons: our computational budget allows only for m P(A|θ ) computations. Then, we determine the range {θ 1 , . . , θ m } of ˜ for the selected m.