306 M.K. Vakilzadeh et al. Algorithm 1 Pseudo code for the systematic resampling scheme at simulation level j Inputs f k j 1, k D1; : : : ; Ntg j 1. / fNwk j 1, k D1; : : : ; Ntg (the importance weights corresponding to k j 1’s ), Algorithm Permute randomly the samples f k j 1 W k D1; : : : ; Ntg, Sample u1 UŒ0;1=Nt , Compute uk D k 1 Nt Cu1 for k D2; : : : ; Nt, for k D1; : : : ; Nt do Set Q k j D l j 1 such that indexl satisfies (30.10), end for Outputs f Q k j , k D1; : : : ; Ntg j. / by drawing a random number ui from a uniform distribution UŒ0;1 . The sample Q i j D l j 1 is then selected such that l satisfies: l 1 XkD1 Nwk j 1 ui l XkD1 Nwk j 1; (30.10) where l is a dummy index. Repeating this procedure for i D1; : : : ; Nt generates a new set of samples f Q i j; i D1; : : : ; Ntg that are approximately distributed according to O j. /. The difference between these resampling schemes arises from the way they generate the sequence of random numbers fui; i D1; : : : ; Ntg. The multinomial resampling scheme, as the most straightforward scheme, draws Nt independent random numbers ui UŒ0;1 , for i D1; : : : ; Nt [19]. On the other hand, the systematic resampling partitions the interval .0;1 into Nt disjoint sets, .0; 1 D.0; 1=Nt [ [..Nt 1/=Nt;1 , and draws the first random number u1 UŒ0;1=Nt . Then, the other random numbers are deterministically linked tou1 as: uk D .k 1/ Nt C u1 fork D2; : : : ; Nt (30.11) Therefore, the outcome of the systematic resampling scheme is deterministically dependent on the order in which samples f k j 1; k D 1; : : : ; Ntg are sorted. To remove such deterministic link, one can randomly permute the samples before resampling [18]. See Algorithm 1 for a detailed implementation of the systematic resampling scheme, which is included to complete the series of pseudo codes for the proposed sequential MCMC algorithm. The resampling schemes introduce a variability to the duplication count Qnk j of samples [21]. To reduce this, in this study we propose to employ the systematic resampling scheme within the described sequential algorithm. 30.1.3 MCMC Sampling The importance resampling step generates a set of samples f Q k j ; k D 1; : : : ; Ntg approximately distributed according to j. / [14]. Assume that this set includes N .0/ j distinct samples f .0/;k j ; k D1; : : : ; N .0/ j g, each duplicated n .0/;k j times. In the move step, an MCMC chain of length n.0/;k j is initiated from each .0/;k j to draw new samples from the target distribution j. /. Because of this we hereafter call the .0/;k j s seeds for MCMC chains. This leads to an enriched set of Nt samples f k j ; k D1; : : : ; Ntg from the support of the intermediate distribution j. /. The role of MCMC sampling in the proposed sequential algorithm is thus to hinder the accumulation of the importance weights on just a few samples as the simulation level j increases [13] and, as a result, helps to avoid the sample impoverishment. The challenge for MCMC algorithms is to devise a proposal PDF that is a good approximation of the posterior distribution and at the same time is inexpensive to manipulate. In this section, we first present a new MCMC algorithm that exploits the local gradient and Hessian information of the negative log posterior to accelerate the MCMC sampling. Afterwards, by exploiting the special structure of the likelihood function (30.2), we propose an augmented Gauss-Newton approximation of the Hessian.
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