Particle Gibbs with backward simulation

Use the density, support, and positive-denominator hypotheses of Backward-simulation particle Gibbs extended target. Select multinomial resampling with ancestor indices in sampled order. Fix $N\ge 2$. Before updating, require the retained parameter-path-index triple $(\theta ,x_{0:T}^j,j)$ to be in $D_{\mathrm{B}\mathrm{S}}$, the set fixed in the Conditional-version convention of Backward-simulation particle Gibbs extended target; otherwise return the unchanged extended state. If a required ratio is undefined or nonfinite, or a required denominator or normalizing sum is zero or nonfinite during a sweep, return the old retained state. A finite zero density ratio is valid: assign that particle zero weight and continue. The factorization theorem proves that the stated failures have conditional probability zero on the designated full-measure set.

A backward-simulation sweep takes the current extended state $(\theta ,U,j)$, where $j=(j_0,\dots ,j_T)$ selects the reference path $x_{0:T}^{\ast }=x_{0:T}^j$ from $U$. It returns the updated extended state $({\theta }^{\mathrm{\prime }},U^{\mathrm{\prime }},J)$ and selected path $x_{0:T}^J$. First draw ${\theta }^{\mathrm{\prime }}$ from its exact posterior full conditional given $x_{0:T}^{\ast }$ and the observations. If $({\theta }^{\mathrm{\prime }},x^{\ast },j)$ is outside $D_{\mathrm{B}\mathrm{S}}$, return the old extended state.

The initial proposal, non-reference resampling and propagation, and SIR weight updates use Conditional particle filter for particle Gibbs in its density setting. Because $j$ is a backward index sequence rather than an ancestral lineage, the reference particle's ancestor is sampled from $B$ below instead of being fixed to the preceding reference index.

1. Initialize the conditional auxiliary particle filter at time $0$: set $X_0^{j_0}=x_0^{\ast }$ and independently draw $X_0^i\sim q_{{\theta }^{\mathrm{\prime }},0}$ for $i\mathrm{\ne }{j}_0$. For each $i$, compute the SIR initial weight $a_0^i={\mu }_{{\theta }^{\mathrm{\prime }},0}(X_0^i)G_0(X_0^i)\mathrm{/}{q}_{{\theta }^{\mathrm{\prime }},0}(X_0^i)$. If ${\sum }_i{a}_0^i>0$, set $W_0^i=a_0^i\mathrm{/}{\sum }_{\mathrm{\ell }}{a}_0^{\mathrm{\ell }}$; otherwise use the SIR zero-weight convention $W_0^i=1\mathrm{/}N$.
2. For $t=1,\dots ,T$, do the following in order:
   1. For each $i\mathrm{\ne }{j}_t$, draw $A_t^i$ categorically with probabilities $W_{t-1}^{1:N}$ and then draw $X_t^i\sim q_{{\theta }^{\mathrm{\prime }},t}(\cdot \mid X_{t-1}^{A_t^i})$.
   2. Set $X_t^{j_t}=x_t^{\ast }$ and draw its ancestor $A_t^{j_t}$ categorically with probabilities $B_{t-1}^{1:N}(x_t^{\ast };U^{\mathrm{\prime }})$.
   3. For every $i$, compute $a_t^i=G_t(X_t^i)f_{{\theta }^{\mathrm{\prime }},t}(X_t^i\mid X_{t-1}^{A_t^i})\mathrm{/}{q}_{{\theta }^{\mathrm{\prime }},t}(X_t^i\mid X_{t-1}^{A_t^i})$. If ${\sum }_i{a}_t^i>0$, set $W_t^i=a_t^i\mathrm{/}{\sum }_{\mathrm{\ell }}{a}_t^{\mathrm{\ell }}$; otherwise use the SIR zero-weight convention $W_t^i=1\mathrm{/}N$.
3. Draw $J_T$ categorically with probabilities $W_T^{1:N}$. For $t=T-1,T-2,\dots ,0$, draw $J_t$ with probabilities $B_t^{1:N}(X_{t+1}^{J_{t+1}};U^{\mathrm{\prime }})$. Return the updated extended state and selected path.

The zero-weight conventions above define normalized weights as in Sequential importance resampling for Feynman–Kac path targets. Retain zero individual density ratios, importance weights, and backward probabilities. If a required ratio is undefined or nonfinite, a backward-probability denominator is zero or nonfinite, or a weight sum is zero or nonfinite, return the old extended state as specified above. On the designated good set these exceptional events have conditional probability zero. For $T=0$, Step 2 is empty and Step 3 draws $J_0$ from $W_0$.

The conditional auxiliary particle filter and the backward simulator are the two particle-coordinate conditionals of Backward-simulation particle Gibbs extended target. Their Gibbs interpretation and exact posterior marginal are separate proved properties.