Particle MCMC: PMMH and particle Gibbs

Fix observations $y_{1:T}$, a standard-Borel parameter space $(\mathrm{\Theta },\mathcal{T})$, a prior probability ${\pi }_0$, and the finite-horizon Feynman–Kac path model in Feynman–Kac path targets and normalizers. Write $L_{\theta }(y_{1:T})={\gamma }_{\theta ,T}(1),C={\int }_{\mathrm{\Theta }}{L}_{\theta }(y_{1:T}){\pi }_0(d\theta )\mathrm{.}$ Require every prefix normalizer $0<Z_{\theta ,t}<\mathrm{\infty }$ for $t=0,\dots ,T$ and ${\pi }_0$-almost every $\theta$, together with $0<C<\mathrm{\infty }$; then the parameter posterior is ${\pi }_y(d\theta )=C^{-1}{L}_{\theta }(y_{1:T}){\pi }_0(d\theta )\mathrm{.}$

For fixed particle count $N\ge 1$, let $U$ contain the full output of Sequential importance resampling for Feynman–Kac path targets at $\theta$ and a terminal index $K\in \{1,\dots ,N\}$ drawn conditionally with its final normalized weight. Let $m_{\theta }(du)$ be this measurable auxiliary-variable probability kernel and let ${\widehat{L}}_{\theta }(U)={\widehat{Z}}_T$ be jointly measurable and nonnegative in $(\theta ,U)$. The required interface is $\int {\widehat{L}}_{\theta }(u)m_{\theta }(du)=L_{\theta }(y_{1:T});$ The sequential particle estimator is unbiased for Feynman–Kac path integrals supplies this identity under its finite-horizon support and integrability hypotheses. Define the extended target probability by $\tilde{\pi }(d\theta ,du)=C^{-1}{\pi }_0(d\theta )m_{\theta }(du){\widehat{L}}_{\theta }(u)\mathrm{.}$

Given current $(\theta ,U)$, propose ${\theta }^{\mathrm{\prime }}\sim R(\theta ,\cdot )$ and then $U^{\mathrm{\prime }}\sim m_{{\theta }^{\mathrm{\prime }}}$. Apply the general Metropolis–Hastings construction Metropolis–Hastings kernel to this proposal kernel and target $\tilde{\pi }$. On rejection retain the entire current pair $(\theta ,U)$, including its stored likelihood estimate and selected path index. When ${\pi }_0$ and $R$ have densities $p_0$ and $r$, and the current extended target density is positive, the acceptance probability is $1\wedge \frac{p_0({\theta }^{\mathrm{\prime }}){\widehat{L}}_{{\theta }^{\mathrm{\prime }}}(U^{\mathrm{\prime }})r(\theta \mid {\theta }^{\mathrm{\prime }})}{p_0(\theta ){\widehat{L}}_{\theta }(U)r({\theta }^{\mathrm{\prime }}\mid \theta )}\mathrm{.}$ For the displayed density-ratio formula, initialize at a current state where $p_0(\theta ){\widehat{L}}_{\theta }(U)>0$. The density-free version uses the accepted-flow Radon–Nikodym construction in the cited general-state algorithm; it does not evaluate a pointwise prior mass or require this density condition. Proposed zero estimates are rejected. The input is the current extended state $(\theta ,U)$; output the proposed extended state $({\theta }^{\mathrm{\prime }},U^{\mathrm{\prime }})$ on acceptance and retain $(\theta ,U)$ on rejection.

The selected path stored in $U$ is part of the auxiliary state. Refreshing its estimate after rejection defines the different Monte Carlo within Metropolis procedure, not this algorithm.

Under the hypotheses of Particle marginal Metropolis–Hastings, assume the unbiasedness identity there holds and the evidence $C$ is finite and positive. Then $\tilde{\pi }$ is a probability measure whose parameter marginal is ${\pi }_y$. With the terminal index included in $U$, the joint marginal of $(\theta ,X_{0:T}^K)$ is the posterior $C^{-1}{\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}_{0:T})$. The PMMH transition kernel is reversible, hence invariant, for $\tilde{\pi }$; its parameter and selected-path marginals are therefore exact for every fixed $N\ge 1$. This is an invariance result only; it asserts no irreducibility, uniqueness, convergence rate, or mixing speed.

Use the parameter and likelihood-estimator inputs of Particle marginal Metropolis–Hastings, restricted to the common-density case where the prior has density $p_0$ and the proposal kernel has transition density $r({\theta }^{\mathrm{\prime }}\mid \theta )$ with respect to a common parameter reference measure. Initialize at a parameter with $p_0(\theta )>0$, and assume ${\widehat{L}}_{\vartheta }(U)>0$ for $m_{\vartheta }$-almost every $U$ at every parameter $\vartheta$ that the procedure can propose or visit. These conditions make the displayed ratio defined almost surely. The input is the current parameter $\theta$; the output is the next parameter.

At each iteration draw a fresh $U\sim m_{\theta }$, propose ${\theta }^{\mathrm{\prime }}\sim R(\theta ,\cdot )$, and independently draw $U^{\mathrm{\prime }}\sim m_{{\theta }^{\mathrm{\prime }}}$. Accept ${\theta }^{\mathrm{\prime }}$ with probability $1\wedge \frac{p_0({\theta }^{\mathrm{\prime }}){\widehat{L}}_{{\theta }^{\mathrm{\prime }}}(U^{\mathrm{\prime }})r(\theta \mid {\theta }^{\mathrm{\prime }})}{p_0(\theta ){\widehat{L}}_{\theta }(U)r({\theta }^{\mathrm{\prime }}\mid \theta )}\mathrm{.}$ Output ${\theta }^{\mathrm{\prime }}$ on acceptance and $\theta$ on rejection. Discard both auxiliary values before the next iteration and draw a new current estimate.

This is Monte Carlo within Metropolis (MCWM). It does not retain a single extended state on rejection, so the pseudo-marginal invariance proof does not apply. This definition makes no exactness or stationarity claim.

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.

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.

Assume the hypotheses of Particle Gibbs extended target and Sequential importance resampling for Feynman–Kac path targets, with multinomial resampling in sampled order, including the finite-horizon support conditions and $0<L_{\theta }<\mathrm{\infty }$ for prior-almost every $\theta$, with $0<C=\int L_{\theta }{\pi }_0(d\theta )<\mathrm{\infty }$. Then ${\phi }_N$ in Particle Gibbs extended target is a probability measure and its selected $(\theta ,X_{0:T}^K)$ marginal is $C^{-1}{\pi }_0(d\theta ){\gamma }_{\theta ,T}(d{x}_{0:T}),$ the exact joint parameter and smoothing posterior, for every $N\ge 2$.

Under Particle Gibbs extended target and Conditional particle filter for particle Gibbs, for ${\phi }_N$-almost every extended state the conditional law of the terminal index $K$ given $(\theta ,U)$ is $\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}(W_T^{1:N})$. For posterior-almost every retained parameter-trajectory-lineage value, the remaining particles and ancestor indices have exactly the conditional-particle-filter law in Conditional particle filter for particle Gibbs. In a joint sweep, the parameter draw from its posterior full conditional given the selected path, followed by regeneration of the remaining particles and ancestors from the conditional-particle-filter law, is the joint conditional-block update of those variables given the retained selected path and lineage. The terminal index is then drawn from its categorical conditional given the particle system. These two updates preserve the extended target; the measurable exceptional-state identity fallback specified by the algorithm is included. The selected marginal is the exact joint parameter-path posterior. This invariance does not assert fast mixing.

Assume the finite-horizon Feynman–Kac model and proposal support conditions in Feynman–Kac path targets and normalizers and Feynman–Kac path Radon–Nikodym weight. Use the multinomial SIR law with ancestor indices in sampled order. Residual resampling is not included in this labeled-density construction. For this variant, additionally assume that the initial target law has a measurable density ${\mu }_{\theta ,0}$, each target transition $M_{\theta ,t}$ has a jointly measurable density $f_{\theta ,t}(x_t\mid x_{t-1})$, and each proposal transition $R_{\theta ,t}$ has a density $q_{\theta ,t}(x_t\mid x_{t-1})$, all with respect to declared sigma-finite coordinate reference measures. Let $q_{\theta ,0}$ be the proposal initial density with respect to the initial coordinate reference measure. Retain the proper-prior and positive-finite-evidence assumptions of Particle Gibbs extended target. Then the path measure has density ${\gamma }_{\theta ,T}(x_{0:T})={\mu }_{\theta ,0}(x_0)G_0(x_0){\prod }_{t=1}^T{f}_{\theta ,t}(x_t\mid x_{t-1})G_t(x_t)\mathrm{.}$ Require positive finite denominators in the backward probabilities below on the extended-target support.

Write $U$ for all SIR particles and ancestor indices, ${\psi }_{\theta }(u)$ for their forward joint density, and $j=(j_0,\dots ,j_T)$ for a sequence of particle indices. Let $x_{0:T}^j$ be the path formed from states $X_t^{j_t}$, without requiring $A_t^{j_t}=j_{t-1}$. For $t=0,\dots ,T-1$, define $B_t^i(z;U)=\frac{W_t^i{f}_{\theta ,t+1}(z\mid X_t^i)}{{\sum }_{\mathrm{\ell }=1}^N{W}_t^{\mathrm{\ell }}{f}_{\theta ,t+1}(z\mid X_t^{\mathrm{\ell }})}\mathrm{.}$ The backward-simulation extended target is ${\phi }_N^{\mathrm{B}\mathrm{S}}(d\theta ,du,j)=\frac{{\pi }_0(d\theta ){\gamma }_{\theta ,T}(x_{0:T}^j)}{C{N}^{T+1}}\frac{{\psi }_{\theta }(u)}{q_{\theta ,0}(X_0^{j_0}){\prod }_{t=1}^T{W}_{t-1}^{A_t^{j_t}}{q}_{\theta ,t}(X_t^{j_t}\mid X_{t-1}^{A_t^{j_t}})}{\prod }_{t=1}^T{B}_{t-1}^{A_t^{j_t}}(X_t^{j_t};U)\mathrm{.}$ where the displayed expression is a density with respect to the common particle reference measures, counting measure on ancestor and selected indices, and the prior measure on $\theta$. The complete SIR forward density ${\psi }_{\theta }$ includes the normalized ancestor probabilities and proposal densities for every particle. This is the backward-simulation extension of the particle Gibbs target; its exact selected-path marginal and Gibbs conditionals are proved in The backward-simulation sweep is a Gibbs update ↗.

Interpret all displayed ratios on their common positive finite denominator sets and set the density to zero elsewhere. The theorem below proves these versions agree with the extended target and its conditional laws almost everywhere. By Disintegration of a joint law on standard borel spaces, fix a common measurable full-posterior-measure good set $D_{\mathrm{B}\mathrm{S}}$ of retained parameter-path-index values on which the normalized conditional auxiliary filter and backward-index formulas are valid. This also excludes exceptional reference values of the chosen density/Radon–Nikodym versions. Off this set, the sweep is the identity on the retained state. A reference value being in topological support is not a replacement for this almost-everywhere convention.

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.

Under Backward-simulation particle Gibbs extended target and Particle Gibbs with backward simulation, for posterior-almost every retained state in the stated good set, the conditional law of the remaining particle system given $(\theta ,x^{\ast },j)$ is the stated conditional auxiliary particle filter, and the conditional law of the index sequence given $(\theta ,U)$ is the backward simulator with terminal weights $W_T$ and backward probabilities $B_t$. In a joint sweep, the posterior draw of ${\theta }^{\mathrm{\prime }}$ given the selected path, followed by conditional auxiliary-particle filter regeneration, is the joint conditional-block update of the parameter and remaining particle variables given the retained path and index sequence. The backward-index draw is then its conditional update given $({\theta }^{\mathrm{\prime }},U^{\mathrm{\prime }})$. The extended target is a probability measure whose selected $(\theta ,x^{\ast })$ marginal is the exact joint smoothing posterior. With the measurable identity fallback on exceptional retained states, these updates preserve the extended target for every fixed $N\ge 2$. No fast-mixing or ergodicity claim follows.