The backward-simulation sweep is a Gibbs update

Write $w_t^i$ for the unnormalized importance weight of particle $i$ using its actual ancestor $A_t^i$. On the proposal-support set in [F3], $q_{\theta ,0}(X_0^i)w_0^i={\mu }_{\theta ,0}(X_0^i)G_0(X_0^i)$ and $q_{\theta ,t}(X_t^i\mid X_{t-1}^{A_t^i})w_t^i=G_t(X_t^i)f_{\theta ,t}(X_t^i\mid X_{t-1}^{A_t^i})$. These product identities include zero weights and zero transition densities; they require no division by an alternative-parent transition density. Put $S_t={\sum }_i{w}_t^i$ and $H_t(z)={\sum }_i{w}_{t-1}^i{f}_{\theta ,t}(z\mid X_{t-1}^i)$. Whenever the prescribed normalizers are positive and finite, $W_t^i=w_t^i\mathrm{/}{S}_t$ and $B_{t-1}^i(z;U)=w_{t-1}^i{f}_{\theta ,t}(z\mid X_{t-1}^i)\mathrm{/}{H}_t(z)$.

Fix a backward index sequence $j$ and its reference path. Expanding the forward density ${\psi }_{\theta }$ in [F1] and canceling the selected initial proposal and each selected joint ancestor-proposal factor gives ${\phi }_N^{\mathrm{B}\mathrm{S}}(d\theta ,du,j)=\frac{{\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}_{0:T}^j)}{C{N}^{T+1}}{\prod }_{i\mathrm{\ne }{j}_0}{q}_{\theta ,0}(x_0^i){\prod }_{t=1}^T\left[{\prod }_{i\mathrm{\ne }{j}_t}(W_{t-1}^{A_t^i}{q}_{\theta ,t}(x_t^i\mid x_{t-1}^{A_t^i}))\right]{\prod }_{t=1}^T{B}_{t-1}^{A_t^{j_t}}(x_t^{j_t};U)$. The bracketed factors are exactly the non-reference resample-and-propagate law and the categorical backward ancestor law in [F2]. For target-almost every retained path, its initial target factor and all its potentials are positive. The reference initial weight is therefore positive on the supplied proposal-support set. Inductively, a reference ancestor drawn from $B$ has positive preceding weight and transition density, so its new importance weight is positive there as well. Thus each weight sum remains positive; zero individual free-particle weights cause no failure. The prescribed support and finite-denominator conventions exclude the remaining exceptional ratios almost everywhere. Each categorical or proposal factor then integrates to one, so this is the normalized conditional auxiliary particle filter.

Fix $(\theta ,U)$ on its forward-support set with positive finite required normalizers. In [F1], cancel each selected ancestor probability against the numerator of its backward factor, then use the product identities in step 1.1 to absorb each potential into the actual-parent weight. The extended density becomes $\frac{{\pi }_0(d\theta ){\psi }_{\theta }(u)}{C{N}^{T+1}}{w}_0^{j_0}{\prod }_{t=1}^T\frac{w_t^{j_t}{f}_{\theta ,t}(X_t^{j_t}\mid X_{t-1}^{j_{t-1}})}{H_t(X_t^{j_t})\mathrm{/}{S}_{t-1}}=\frac{{\pi }_0(d\theta ){\psi }_{\theta }(u)}{C{N}^{T+1}}\left({\prod }_{t=0}^T{S}_t\right)W_T^{j_T}{\prod }_{t=1}^T{B}_{t-1}^{j_{t-1}}(X_t^{j_t};U)$. This equality merely regroups products; no weight or transition density is divided out. A zero selected weight or path-transition factor gives zero mass on both sides. Index sequences encountering a zero backward denominator likewise have zero mass: a positive terminal weight has a positive actual-parent factor, and a positive backward draw recursively has positive preceding weight. Set these zero-mass terms to zero as prescribed by [F1]. The prefactor is independent of $j$, and summing the remaining factors from $j_0$ through $j_T$ gives one by their categorical normalization. Thus the index conditional is exactly the backward simulator in [F2], including $T=0$, when the product is empty.

By step 1.2, for each fixed $j$ the other particle variables have a normalized conditional law. The leading factor is $C^{-1}{\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}^j)\mathrm{/}{N}^{T+1}$ by [F4]. There are $N^{T+1}$ index sequences, so integrating the remaining variables and summing over them gives total mass one and selected $(\theta ,x^{\ast })$ marginal $C^{-1}{\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}^{\ast })$.

The standard-Borel state and particle spaces permit [F5] to be applied to the probability target in step 2.2. The normalized factorizations in steps 1.2 and 2.1 identify its conditional kernels outside measurable marginal-null sets. Pull exceptional events back to the common extended-state space and take their finite union over index sequences and sweep blocks; its target mass is zero. By the conditional integral identity in [F5], its conditional probability given the retained variables is zero outside a measurable marginal-null set. Removing this set gives the good set specified in [F1]–[F2]. On its complement the algorithm retains the old state; undefined ratio or normalization events within a conditional factorization have zero conditional mass on this good set. Thus the stated formulas and their measurable fallback define conditional-law versions almost everywhere, without assuming arbitrary reference points admit valid density ratios.

In coordinates consisting of the selected path $x^j$, its index sequence $j$, and the remaining particle variables $U_{-j}$, the factorization in step 1.2 shows that the conditional law of $(\theta ,U_{-j})$ given $(x^j,j)$ is obtained by first drawing $\theta$ from its posterior full conditional given $x^j$ and then drawing $U_{-j}$ from the conditional auxiliary-particle-filter law. The factor $N^{-(T+1)}$ is constant in $\theta$ and $x^j$, so the old particle system has been marginalized in this joint conditional block. Thus the first two updates in the algorithm, parameter draw and filter regeneration, together preserve the extended target. By step 2.1, the following backward-index draw is the conditional law of $j$ given $(\theta ,U)$ and also preserves that target. The exceptional-state identity modifications do not change its integral by step 3.1, and the selected marginal in step 2.2 remains exact.