Slide: Comparison Direct Method vs FB and TF Assume we are interested in approximating R T = T=1  (xt ) p ( xt j y1:T ) dxt using SMC. t Fixed-lag N O (TN ) O (T /N ) T T 2 2 +O (T /N ) Direct SMC N O (TN ) O T 2 /N O (T /N ) O T 2 /N  Method # particles cost Var. Bias MSE=Bias2 +Var  FB/TF N O TN 2 , O (TN ) O (T /N ) O (T /N ) O T 2 /N 2  Naive implementations FB and TF have MSE of same order as direct method for xed computational complexity but MSE is bias dominated for FB/TF whereas it is variance dominated for Direct SMC. Fast implementations FB and TF of computational complexity O (NT ) outperform other approaches as MSE is O T 2 /N 2 whereas it is O T 2 /N for direct SMC. A. Doucet (MLSS Sept. 2012) Sept. 2012 74 / 136
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