Conditional particle filter for particle Gibbs

Fix $N\ge 2$, a finite horizon $T$, a parameter $\theta$, and a reference trajectory $x_{0:T}^{\ast }$ with a reference lineage of particle indices $b_0,\dots ,b_T\in \{1,\dots ,N\}$. Use the proposal kernels, incremental weights, and normalized resampling weights of Sequential importance resampling for Feynman–Kac path targets. Use multinomial resampling with indices in sampled order. Require target-path domination as in Feynman–Kac path Radon–Nikodym weight and a retained $(\theta ,x^{\ast },b)$ in the measurable full-measure good set $D$ of Particle Gibbs extended target. The formulas are conditional-law versions for posterior-almost every reference path; membership in topological support alone is insufficient. Outside $D$, return status “unavailable” and leave the retained parameter-path-lineage state unchanged in the sweep. If an arithmetic operation in a sampled particle system is undefined or a required normalizing sum is zero/nonfinite, also return “unavailable” and retain the old state; on $D$ these events have conditional probability zero by the factorization theorem.

At fixed $\theta$, use the parameter-specific proposal kernels and weights in Sequential importance resampling for Feynman–Kac path targets, writing its normalized resampling weights ${\bar{w}}_t^i$ as $W_t^i$. The input is $(\theta ,x_{0:T}^{\ast },b_{0:T})$; the output is the conditional particle system, its normalized weights, its ancestry arrays, and either a successful status or “unavailable”.

1. At time $0$, draw each $X_0^i$ for $i\mathrm{\ne }{b}_0$ independently from ${\nu }_0$ and set $X_0^{b_0}=x_0^{\ast }$. Compute the SIR initial weights $a_0^i=G_0(X_0^i){\rho }_0(X_0^i)$ and set ${\widehat{Z}}_0=N^{-1}{\sum }_i{a}_0^i$. If ${\sum }_i{a}_0^i>0$, put $W_0^i=a_0^i\mathrm{/}{\sum }_j{a}_0^j$; otherwise use the SIR zero-weight convention $W_0^i=1\mathrm{/}N$. A zero or nonfinite normalizing sum makes the sweep unavailable and returns the unchanged retained state.
2. For $t=1,\dots ,T$, for every $i\mathrm{\ne }{b}_t$ draw an ancestor $A_t^i$ from the categorical distribution with probabilities $W_{t-1}^{1:N}$ and then draw $X_t^i$ from the proposal kernel conditional on $X_{t-1}^{A_t^i}$. Set $A_t^{b_t}=b_{t-1}$ and $X_t^{b_t}=x_t^{\ast }$. Compute the SIR incremental weights $a_t^i=G_t(X_t^i)r_t(X_{t-1}^{A_t^i},X_t^i)$ and update ${\widehat{Z}}_t={\widehat{Z}}_{t-1}(N^{-1}{\sum }_i{a}_t^i)$. If the weight sum is positive, put $W_t^i=a_t^i\mathrm{/}{\sum }_j{a}_t^j$; otherwise use the SIR zero-weight convention $W_t^i=1\mathrm{/}N$. A zero or nonfinite normalizing sum makes the sweep unavailable and returns the unchanged retained state.
3. Return the conditional particle system, its normalized weights, and its ancestry arrays.

For a joint parameter-path sweep, retain the current selected path $x^{\ast }$ and its lineage $b$, draw ${\theta }^{\mathrm{\prime }}$ from its exact posterior full conditional given $x^{\ast }$, and apply Steps 1–3 at ${\theta }^{\mathrm{\prime }}$. This parameter draw followed by conditional-particle-filter regeneration is one joint conditional-block update. Then draw $K^{\mathrm{\prime }}\sim \mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}(W_T^{1:N})$ and trace its ancestry to obtain the next selected path. Output the new extended state $({\theta }^{\mathrm{\prime }},U^{\mathrm{\prime }},K^{\mathrm{\prime }})$ together with that path. The correctness of this sweep is a separate result.