Particle Gibbs extended target

Let $(\mathrm{\Theta },\mathcal{T})$ be a standard-Borel parameter space with a prior ${\pi }_0$. Use common standard-Borel state spaces and a jointly measurable parameterized target/proposal family: the initial and transition target and proposal laws are probability kernels in $\theta$ and their state arguments, the potentials are jointly measurable, and the chosen Radon–Nikodym weight versions are jointly measurable. Thus $\theta \mapsto {\gamma }_{\theta ,T}$ and $\theta \mapsto {\psi }_{\theta }$ are measure and probability kernels, and the SIR weights and normalizer are jointly measurable in $(\theta ,U)$. The prior-kernel law ${\pi }_0(d\theta ){\psi }_{\theta }(du)$ is a probability measure by A prior and model kernel define a joint law.

Use the finite-horizon state path model and proposal support conditions in Feynman–Kac path targets and normalizers, Feynman–Kac path Radon–Nikodym weight, and Sequential importance resampling for Feynman–Kac path targets. Select the multinomial resampling branch, returning ancestor indices in sampled order. Residual resampling is not the labeled particle law used by this genealogical construction. Write ${\gamma }_{\theta ,T}$ for the Feynman–Kac path measure at parameter $\theta$ and set $L_{\theta }={\gamma }_{\theta ,T}(1)$. Assume $0<L_{\theta }<\mathrm{\infty }$ for ${\pi }_0$-almost every $\theta$, and $C=\int L_{\theta }{\pi }_0(d\theta ),0<C<\mathrm{\infty }\mathrm{.}$ For a fixed particle count $N\ge 2$, write $U=(X_0^{1:N},A_1^{1:N},X_1^{1:N},\dots ,A_T^{1:N},X_T^{1:N})$ for the full particle system and ancestor indices generated by Sequential importance resampling for Feynman–Kac path targets. Its forward probability law at parameter $\theta$ is ${\psi }_{\theta }(du)$. The algorithm defines the product normalizer estimate ${\widehat{Z}}_T(U)$ and final normalized weights $W_T^1(U),\dots ,W_T^N(U)$; $K$ is a terminal particle index. The path $X_{0:T}^K$ means the states on the ancestral lineage of particle $K$, traced through the arrays $A_1^{1:N},\dots ,A_T^{1:N}$.

More explicitly, relative to counting measure on ancestor arrays and the ordered SIR proposal-kernel integrals for the states, ${\psi }_{\theta }(du)={\prod }_{i=1}^N{\nu }_{\theta ,0}(d{x}_0^i){\prod }_{t=1}^T{\prod }_{i=1}^N[W_{t-1}^{A_t^i}(U)R_{\theta ,t}(X_{t-1}^{A_t^i},d{x}_t^i)]\mathrm{.}$ Each factor $W_{t-1}^{A_t^i}(U)$ is the categorical mass of that ancestor choice; no state transition density is assumed for the genealogical variant.

The genealogical particle Gibbs extended target is the measure ${\phi }_N(d\theta ,du,k)=C^{-1}{\pi }_0(d\theta ){\psi }_{\theta }(du){\widehat{Z}}_T(u)W_T^k(u),k\in \{1,\dots ,N\}\mathrm{.}$ The selected state is $(\theta ,X_{0:T}^K)$. The zero-weight fallback in the particle algorithm defines $W_T$ everywhere; when ${\widehat{Z}}_T=0$ the extended target assigns the configuration zero mass.

The measure is a probability and has the exact joint smoothing posterior as its selected $(\theta ,X_{0:T}^K)$ marginal by The particle Gibbs extended target has the exact smoothing marginal ↗.

All particle variables, indices, parameter and retained-path coordinates are standard Borel. Under AC, Disintegration of a joint law on standard borel spaces supplies everywhere probability-kernel conditional versions of this extended target. The explicit conditional-filter formulas below identify those versions only outside a measurable null set of retained parameter-path-lineage values. Fix a measurable full-measure set $D$ on which the formulas are defined and normalized, intersected over the finitely many lineages and conditional updates. Reference paths are admitted through $D$, rather than through an assertion about arbitrary points of topological support. On its complement a particle Gibbs sweep leaves the retained state unchanged. This identity fallback is measurable and changes the kernel only on a posterior-null set. The factorization theorem establishes the required full-measure property; no pointwise validity for arbitrary Radon–Nikodym versions is asserted.