The conditional particle Gibbs sweep is a Gibbs update

Since the only $K$-dependent factor in [F1] is $W_T^K$, and the final normalized weights sum to one, ${\phi }_N(K=k\mid \theta ,U)=W_T^k$.

Fix a selected index $K$ and trace its lineage $b_0,\dots ,b_T$. On the positive-mass support of [F1], write ${\bar{w}}_t=N^{-1}{\sum }_i{w}_t^i$, so ${\widehat{Z}}_T{W}_T^K=({\prod }_{t=0}^{T-1}{\bar{w}}_t)w_T^{b_T}\mathrm{/}N$. In the forward density [F2], the selected lineage contributes ${\prod }_{t=1}^T{W}_{t-1}^{b_{t-1}}{R}_t(x_{t-1}^{b_{t-1}},d{x}_t^{b_t})$ and its initial proposal ${\nu }_0(d{x}_0^{b_0})$. Substituting $W_{t-1}^{b_{t-1}}=w_{t-1}^{b_{t-1}}\mathrm{/}(N{\bar{w}}_{t-1})$ cancels each ${\bar{w}}_{t-1}$, leaving the selected path factor $N^{-(T+1)}{\nu }_0(d{x}_0^{b_0})w_0^{b_0}{\prod }_{t=1}^T{R}_t(x_{t-1}^{b_{t-1}},d{x}_t^{b_t})w_t^{b_t}$. If any average weight is zero, [F1] assigns the configuration zero extended-target mass, so its conditional law need not be specified there.

By the initial and transition Radon–Nikodym weight identities in [F4], the selected path factor in step 2.1 equals $N^{-(T+1)}{\gamma }_{\theta ,T}(d{x}_{0:T}^b)$. All remaining factors are exactly the conditional SIR laws for the non-reference particles and ancestors in [F3]. Thus, conditional on the selected path and lineage, the remaining particle variables have the stated conditional-particle-filter law; the same factorization makes the lineage indices uniform given the selected path.

By [F5], the extended target is a probability on a standard-Borel product. Apply [F6] to the particle system conditional on its retained parameter-path-lineage variables. The normalized factorization in step 3.1 identifies its conditional kernel outside a single measurable marginal-null set; pull their exceptional events back to the common extended-state space and take the finite union over lineages and update blocks. This union has target mass zero. Its conditional probability given the retained parameter-path-lineage is therefore zero outside a measurable marginal-null set, by the defining integral identity in [F6]; remove that set to obtain $D$ in [F3]. The same restriction excludes infinite or zero denominators on the positive extended-target support. On the exceptional set the algorithm uses the identity transition on the retained state, so it remains an everywhere measurable probability kernel without claiming pointwise Radon–Nikodym identities at null reference values.

In the coordinates consisting of the selected path $x^b$, its lineage $b$, and the remaining particle variables $U_{-b}$, the factorization in step 3.1 gives the conditional law of $(\theta ,U_{-b})$ given $(x^b,b)$. Its leading factor is ${\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}^b)\mathrm{/}(C{N}^{T+1})$; the factor $N^{-(T+1)}$ makes $b$ uniform given $x^b$ and independent of $\theta$, while the conditional law of $U_{-b}$ given $(\theta ,x^b,b)$ is exactly the conditional-particle-filter law in [F3]. Thus the posterior draw of $\theta$ given $x^b$, followed by regeneration of $U_{-b}$ with that filter, samples the joint conditional law of $(\theta ,U_{-b})$ given $(x^b,b)$. This is a conditional-block update; the parameter draw alone is not asserted to be conditional on the old full particle system.

The block in step 4.2 is the conditional law of $(\theta ,U_{-b})$ given the retained $(x^b,b)$, so replacing that block preserves ${\phi }_N$ by the defining conditional-probability identity. By step 1.1, the subsequent draw of $K$ given the regenerated particle system is its conditional law $\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}(W_T^{1:N})$ and also preserves ${\phi }_N$. The exceptional-state identity modification changes the target integral only on a target-null set by step 4.1. Therefore their composition preserves ${\phi }_N$, whose selected marginal is the exact posterior by [F5].