Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis–Hastings Method

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. 2022 Jan 12;24(1):115. doi: 10.3390/e24010115

Algorithm 1 Particle Marginal Metropolis–Hastings (PMMH) Method.

1:
initialize the parameters $Θ [0]$
2:
for $k = 1, \dots, K$ (K is the number of samples) do
3:
draw the sample candidate of parameters $Θ^{*} \sim q (Θ^{*} | Θ [k - 1])$
4:
draw the initial particles $z_{1}^{(m)} \sim p (z_{1})$ for $m = 1, \dots, M$ (m is the particle number of the particle that is the source of resampling)
5:
calculate the weights of particles $\{w_{1}^{(1)}, w_{1}^{(2)}, \dots, w_{1}^{(M)}\}$ with Equation (8)
6:
normalize the weights of particles $\{W_{1}^{(1)}, W_{1}^{(2)}, \dots, W_{1}^{(M)}\}$ with Equation (7)
7:
resample the particles $\{z_{1}^{(1)}, z_{1}^{(2)}, \dots, z_{1}^{(M)}\}$ according to the normalized weights $\{W_{1}^{(1)}, W_{1}^{(2)}, \dots, W_{1}^{(M)}\}$
8:
for $n = 2, \dots, N$ do
9:
draw the particles $\{z_{n}^{(1)}, z_{n}^{(2)}, \dots, z_{n}^{(M)}\}$ at time step n with Equation (6)
10:
calculate the weights of particles $\{w_{n}^{(1)}, w_{n}^{(2)}, \dots, w_{n}^{(M)}\}$ with Equation (8)
11:
normalize the weights of particles $\{W_{n}^{(1)}, W_{n}^{(2)}, \dots, W_{n}^{(M)}\}$ with Equation (7)
12:
resample the particles $\{z_{n}^{(1)}, z_{n}^{(2)}, \dots, z_{n}^{(M)}\}$ according to the normalized weights $\{W_{n}^{(1)}, W_{n}^{(2)}, \dots, W_{n}^{(M)}\}$
13:
end for
14:
calculate the marginal likelihood $p (y_{1 : N} | Θ^{*})$ with Equation (9)
15:
calculate the acceptance probability $p_{accept}$ with Equation (3)
16:
draw a uniform random number $α \sim U (0, 1)$ ( $U (a, b)$ is a uniform distribution with range $[a, b)$ )
17:
if $α \leq p_{accept}$ then
18:
set the sample of parameters $Θ [k] \leftarrow Θ^{*}$
19:
else
20:
set the sample of parameters $Θ [k] \leftarrow Θ [k - 1]$
21:
end if
22:
end for
23:
return ${\{Θ [k]\}}_{k = 1}^{K}$