Algorithm 2 Replica Exchange Particle Marginal Metropolis–Hastings (REPMMH) Method.

1:initialize the parameters $\left\{\mathsf{\Theta }\right\}\left[0\right]$ 2:for$k=1,\dots ,K$do 3:   for $r=1,\dots ,R$ do 4:     draw the sample candidate of parameters ${\mathsf{\Theta }}^{\left(r\right)*}\sim q\left({\mathsf{\Theta }}^{\left(r\right)*}\mid {\mathsf{\Theta }}^{\left(r\right)}\left[k-1\right]\right)$ 5:     calculate the marginal likelihood $p\left({\mathit{y}}_{1:N}|{\mathsf{\Theta }}^{\left(r\right)*}\right)$ by using the SMC method according to Equation (13) 6:     calculate the acceptance probability $p_{\mathrm{accept}}^{\left(r\right)}$ with Equation (14) 7:     draw a uniform random number $\alpha \sim \mathcal{U}\left(0,1\right)$ ($\mathcal{U}\left(a,b\right)$ is a uniform distribution with range $[a,b)$) 8:     if $\alpha \le p_{\mathrm{accept}}^{\left(r\right)}$ then 9:        set the sample of parameters ${\mathsf{\Theta }}^{\left(r\right)}\left[k\right]\leftarrow {\mathsf{\Theta }}^{\left(r\right)*}$ 10:     else 11:        set the sample of parameters ${\mathsf{\Theta }}^{\left(r\right)}\left[k\right]\leftarrow {\mathsf{\Theta }}^{\left(r\right)}\left[k-1\right]$ 12:     end if 13:   end for 14:   choose the replica number $r_{\mathrm{EX}}\leftarrow 1$ or $r_{\mathrm{EX}}\leftarrow 2$ for the exchange 15:   repeat 16:     calculate exchange probability $p_{\mathrm{EX}}\left(\left\{\mathsf{\Theta }\right\},{\left\{\mathsf{\Theta }\right\}}^{*}\right)$ with Equation (15) for replica numbers $r_{\mathrm{EX}}$ and $r_{\mathrm{EX}}+1$ 17:     draw a uniform random number ${\alpha }_{\mathrm{EX}}\sim \mathcal{U}\left(0,1\right)$ 18:     if ${\alpha }_{\mathrm{EX}}\le p_{\mathrm{EX}}\left(\left\{\mathsf{\Theta }\right\},{\left\{\mathsf{\Theta }\right\}}^{*}\right)$ then 19:        exchange replicas $\left({\mathsf{\Theta }}^{\left(r_{\mathrm{EX}}\right)}\left[k\right],{\mathsf{\Theta }}^{\left(r_{\mathrm{EX}}+1\right)}\left[k\right]\right)\leftarrow \left({\mathsf{\Theta }}^{\left(r_{\mathrm{EX}}+1\right)}\left[k\right],{\mathsf{\Theta }}^{\left(r_{\mathrm{EX}}\right)}\left[k\right]\right)$ 20:     end if 21:     set the replica number $r_{\mathrm{EX}}\leftarrow r_{\mathrm{EX}}+2$ for the exchange 22:   until $r_{\mathrm{EX}}\le R-1$ 23:end for 24:return${\left\{\left\{\mathsf{\Theta }\right\}\left[k\right]\right\}}_{k=1}^K$