Single-Trajectory Gibbs Sampling Overview

• Single-trajectory Gibbs sampling is a method using one evolving Gibbs or Gibbs-like trajectory to perform inference with block updates and warm-start strategies.
• It applies in diverse areas such as particle Gibbs in state-space models, continuous-space MCMC under log-concavity, sparse Gaussian graphical model learning, and quantum thermal estimation.
• Key insights include rigorous mixing time bounds, enhanced sampling efficiency via ancestor and backward sampling, and innovative marginalization techniques for complex distributions.

Searching arXiv for papers on single-trajectory Gibbs sampling and closely related formulations. Single-trajectory Gibbs sampling denotes a class of procedures in which inference, sampling, or estimation is organized around one evolving Gibbs or Gibbs-like Markov trajectory rather than around repeatedly regenerated approximately independent draws. The phrase is used in several technically distinct settings. In state-space models, it commonly refers to particle Gibbs, where a single reference latent path is retained inside a conditional sequential Monte Carlo sweep and updated as one block (Gauraha, 2020, Chopin et al., 2013). In continuous-space Markov chain Monte Carlo, it includes analyses of the Gibbs sampler itself, also known as the coordinate hit-and-run algorithm, under log-smoothness and strong log-concavity assumptions (Wadia, 2024). In quantum thermal-state estimation, it refers to collecting measurement outcomes along one Gibbs-sampling trajectory after burn-in, exploiting the possibility that autocorrelation times can be substantially shorter than mixing times (Jiang et al., 13 Feb 2026, Chen et al., 23 Mar 2026).

1. Scope and shared structure

Single-trajectory methods share a common operational motif: a Markov kernel leaves a target distribution invariant, but the computational objective is pursued using temporally correlated states from one chain realization. What changes across domains is the object represented by the trajectory. In coordinate Gibbs sampling it is a path in $\mathbb{R}^n$; in particle Gibbs it is a latent state trajectory $x_{1:T}$ or $x_{0:T}$ embedded in an extended target; in Continuous-Time Bayesian Networks it is an entire component trajectory over $[0,T]$ conditioned on the remaining components; and in quantum settings it is a sequence of post-measurement states produced by alternating a Gibbs sampler with detailed-balanced measurement channels (Wadia, 2024, Gauraha, 2020, El-Hay et al., 2012, Jiang et al., 13 Feb 2026).

A recurring technical distinction is between preserving the target exactly and controlling deviation from it. Particle Gibbs and its variants are constructed so that the stationary distribution is exactly the desired posterior despite the use of conditional SMC (Gauraha, 2020, Chopin et al., 2013). The log-concave Euclidean analysis instead gives an explicit total-variation mixing bound for the lazy Gibbs sampler from an $M$-warm start (Wadia, 2024). Quantum single-trajectory protocols typically require an initial burn-in of order the mixing time, after which measurements are inserted in a way that preserves the Gibbs ensemble or yields a warm start for rapid re-mixing (Jiang et al., 13 Feb 2026, Chen et al., 23 Mar 2026). By contrast, learning sparse Gaussian graphical models from a single Glauber trajectory is formulated specifically so that the trajectory-length guarantee does not depend on the mixing time (Shen et al., 30 Jun 2026).

A common misconception is that “single-trajectory” implies “single-site” or “one-coordinate-at-a-time” throughout. In fact, the term covers both local-update chains such as Glauber dynamics and block methods that update an entire latent path at once. Another misconception is that a single trajectory automatically obviates equilibration; the available results are more specific. Warm starts and burn-in remain central in several settings, whereas the no-mixing guarantees in sparse Gaussian graphical model learning rely on specialized local window tests and robust aggregation rather than on a general principle that mixing can always be ignored (Wadia, 2024, Shen et al., 30 Jun 2026).

2. Euclidean Gibbs dynamics and learning from one Glauber trajectory

For continuous targets on $\mathbb{R}^n$, the Gibbs sampler considered in "A mixing time bound for Gibbs sampling from log-smooth log-concave distributions" targets

$\pi(x)\propto \exp(-f(x)),$

where $f:\mathbb{R}^n\to\mathbb{R}$ is twice differentiable, $\mu$-strongly convex, and $L$-smooth, equivalently satisfying

$x_{1:T}$0

with condition number $x_{1:T}$1 (Wadia, 2024). The initial distribution $x_{1:T}$2 is assumed to be $x_{1:T}$3-warm with respect to $x_{1:T}$4, meaning

$x_{1:T}$5

Under these assumptions, if $x_{1:T}$6 is the law of the lazy Gibbs sampler after $x_{1:T}$7 steps and $x_{1:T}$8, there is a universal constant $x_{1:T}$9 such that

$x_{0:T}$0

whenever

$x_{0:T}$1

In soft-$x_{0:T}$2 notation,

$x_{0:T}$3

The proof proceeds through restriction to a high-probability compact region, lower bounds on $x_{0:T}$4-conductance, an axis-disjoint partition argument, an isoperimetric inequality, and the Lovász–Simonovits bound for lazy reversible chains (Wadia, 2024).

A different single-trajectory use arises in learning sparse Gaussian graphical models from one continuous-time Glauber trajectory. There the target distribution is a zero-mean Gaussian graphical model with precision matrix $x_{0:T}$5, graph $x_{0:T}$6 defined by $x_{0:T}$7, and $x_{0:T}$8-sparsity specified by degree at most $x_{0:T}$9 together with

$[0,T]$0

(Shen et al., 30 Jun 2026). The observed data are a single continuous-time single-site Gibbs trajectory up to horizon $[0,T]$1, including update times, updated coordinates, and post-update values. The algorithm first estimates the diagonal entries $[0,T]$2 robustly and rescales to the unit-diagonal case, then applies a local edge test based on short windows with update pattern “iiji,” and finally aggregates the resulting local statistics with a robust median-based estimator. On clean windows, after unit-diagonal normalization, the local statistic satisfies

$[0,T]$3

which isolates pairwise influence despite the temporal dependence of a single trajectory (Shen et al., 30 Jun 2026).

The main structure-learning theorem states that there is a polynomial-time algorithm that recovers the edge set with probability at least $[0,T]$4 provided

$[0,T]$5

A parameter-learning theorem gives $[0,T]$6-multiplicative recovery of the entries under

$[0,T]$7

A mixing-based variant using shorter “iji” patterns yields

$[0,T]$8

but the principal result is explicitly mixing-time free (Shen et al., 30 Jun 2026).

3. Particle Gibbs as single-trajectory block Gibbs sampling

In nonlinear or non-Gaussian state-space models, single-trajectory Gibbs sampling most often refers to particle Gibbs. The latent process is

$[0,T]$9

with prior $M$0 and joint posterior

$M$1

The ideal Gibbs sampler would alternate

$M$2

but the path draw is generally intractable (Gauraha, 2020).

Particle Gibbs replaces the path update by a conditional SMC kernel $M$3 that leaves the exact conditional posterior invariant:

$M$4

Operationally, the previous Gibbs iterate $M$5 is used as a reference trajectory. One particle is clamped to this path, the remaining $M$6 particles are propagated and resampled, and at time $M$7 one of the $M$8 complete trajectories is selected according to normalized final weights (Gauraha, 2020). With proposal

$M$9

the unnormalized importance weights are

$\mathbb{R}^n$0

In the bootstrap filter, $\mathbb{R}^n$1, so $\mathbb{R}^n$2 (Gauraha, 2020).

The central single-trajectory feature is the block update of the entire latent path. This contrasts with one-state-at-a-time Gibbs and is explicitly motivated by improved mixing in high-dimensional path spaces. Each conditional SMC sweep costs $\mathbb{R}^n$3, so a particle Gibbs run of $\mathbb{R}^n$4 MCMC iterations costs $\mathbb{R}^n$5. The tutorial treatment notes that good mixing often requires $\mathbb{R}^n$6, giving $\mathbb{R}^n$7 per iteration in long sequences; this is a principal reason for subsequent refinements such as ancestor sampling, blocking, interacting PMCMC, and marginalized constructions (Gauraha, 2020).

4. Extended targets, ancestor sampling, and marginalized variants

The theoretical foundation of particle Gibbs is an extended target distribution on the particle system and ancestry variables. "On particle Gibbs sampling" formulates the target on

$\mathbb{R}^n$8

so that the marginal law of the distinguished trajectory is exactly the Feynman–Kac path target $\mathbb{R}^n$9 (Chopin et al., 2013). A coupling construction between two particle Gibbs updates starting from different trajectories shows that the coupling probability can be made arbitrarily close to one by increasing the number of particles, and a direct corollary is that the particle Gibbs kernel is uniformly ergodic for each fixed $\pi(x)\propto \exp(-f(x)),$0 when $\pi(x)\propto \exp(-f(x)),$1 is large enough. The same work shows that adding a backward-sampling step that resamples the ancestral lineage of the selected particle yields a theoretically more efficient algorithm, in the sense of reduced asymptotic variance, and also extends the method to lower-variance resampling schemes such as residual and systematic resampling (Chopin et al., 2013).

Ancestor sampling and backward sampling are now standard remedies for path degeneracy. In one common formulation, for the reference particle at time $\pi(x)\propto \exp(-f(x)),$2,

$\pi(x)\propto \exp(-f(x)),$3

so the ancestor of the frozen path is no longer fixed deterministically (Gauraha, 2020). The PEIS variant replaces locally designed proposals by approximately fully globally adapted importance densities. It defines proposal kernels

$\pi(x)\propto \exp(-f(x)),$4

through Efficient Importance Sampling regressions and uses auxiliary targets

$\pi(x)\propto \exp(-f(x)),$5

thereby reducing weight variance and improving update rates, effective sample size, and Metropolis–Hastings acceptance behavior in particle-Gibbs-based samplers (Grothe et al., 2016).

A separate line of work improves mixing by analytically eliminating static parameters when conjugacy is available. In "Parameter elimination in particle Gibbs sampling" the complete-data model is assumed to belong to a restricted exponential family,

$\pi(x)\propto \exp(-f(x)),$6

with conjugate prior

$\pi(x)\propto \exp(-f(x)),$7

The resulting marginalized predictive density is available in closed form:

$\pi(x)\propto \exp(-f(x)),$8

Although the induced latent model is non-Markovian, sufficient-statistic updates preserve linear $\pi(x)\propto \exp(-f(x)),$9 complexity. As $f:\mathbb{R}^n\to\mathbb{R}$0, the marginalized samplers approach collapsed Gibbs rather than the unmarginalized Gibbs sampler, which the paper identifies as the mechanism by which performance can improve beyond that of the underlying exact Gibbs scheme (Wigren et al., 2019).

"Particle Gibbs without the Gibbs bit" pushes this logic further by replacing the $f:\mathbb{R}^n\to\mathbb{R}$1 Gibbs step with a marginal joint update. It introduces an augmented density

$f:\mathbb{R}^n\to\mathbb{R}$2

runs one conditional SMC over the induced marginal Feynman–Kac model, and then performs an index swap accepted with a Metropolis–Hastings ratio based on the terminal posterior over the parameter index. The per-iteration complexity is $f:\mathbb{R}^n\to\mathbb{R}$3, and the paper states that one typically takes $f:\mathbb{R}^n\to\mathbb{R}$4, making the cost essentially $f:\mathbb{R}^n\to\mathbb{R}$5 (Corenflos, 7 May 2025).

5. Continuous-time factorized processes

In factorized continuous-time Markov processes, single-trajectory Gibbs sampling appears as exact resampling of one component trajectory conditioned on the others. For a Continuous-Time Bayesian Network with components $f:\mathbb{R}^n\to\mathbb{R}$6, graph-induced parent sets $f:\mathbb{R}^n\to\mathbb{R}$7, and conditional rate matrices $f:\mathbb{R}^n\to\mathbb{R}$8, the posterior target is

$f:\mathbb{R}^n\to\mathbb{R}$9

Exact inference in the full amalgamated rate matrix is exponential in the number of components, so the sampler repeatedly selects one component $\mu$0 and resamples its entire path conditional on the current trajectories of all other components and the evidence (El-Hay et al., 2012).

Conditioned on the remaining trajectories, the selected component remains Markov but becomes time-inhomogeneous. On each subinterval where the rest of the system is constant, $\mu$1 evolves with a piecewise-constant rate matrix $\mu$2 whose off-diagonal entries are inherited from the CTBN conditional intensity matrix:

$\mu$3

Sampling the conditional path is reduced to sampling from this time-inhomogeneous continuous-time Markov process with fixed endpoints. The method computes backward messages $\mu$4 and forward scalars $\mu$5 by matrix-exponential recursions over the jump times of the Markov blanket, constructs the cumulative distribution of the first jump, performs exact “$\mu$6-inversion” to locate the jump time, samples the destination state, and then recurses on the remaining interval (El-Hay et al., 2012).

The significance of this construction is twofold. First, each component update is an exact draw from the correct single-component conditional distribution, so the overall Markov chain has the desired posterior over continuous-time trajectories as stationary distribution. Second, the update exploits graph structure: only the Markov blanket of the resampled component enters the forward-backward calculations. The paper characterizes the procedure as the first that can provide asymptotically unbiased approximation in such processes, with the only approximation arising from finite runtime and numerical tolerance in solving the scalar inversion problem (El-Hay et al., 2012).

6. Quantum thermal-state estimation along one Gibbs trajectory

In quantum many-body settings, single-trajectory Gibbs sampling refers to estimating thermal expectation values from one Gibbs-sampling trajectory rather than preparing the Gibbs state anew between measurements. The target state is

$\mu$7

and the sampler is a CPTP map $\mu$8 satisfying $\mu$9-detailed balance with respect to $L$0 (Jiang et al., 13 Feb 2026). After burn-in of length $L$1, the sampling stage alternates the Gibbs sampler with a coherent measurement channel $L$2, using the composed channel

$L$3

and records the outcome of the first $L$4 in each sandwich. The empirical estimator is

$L$5

The crucial observation is that, if $L$6 also satisfies detailed balance, the measurements do not drive the chain out of equilibrium, and the relevant cost scale becomes the integrated autocorrelation time rather than the full mixing time (Jiang et al., 13 Feb 2026).

The detailed-balance condition is stated through the $L$7-weighted inner product

$L$8

For channels satisfying this symmetry, the spectrum is real in $L$9, and with spectral gap $x_{1:T}$00 one has relaxation time $x_{1:T}$01. The paper proves

$x_{1:T}$02

whereas the mixing time obeys bounds involving the additional factor $x_{1:T}$03 with $x_{1:T}$04 (Jiang et al., 13 Feb 2026). This separates the one-time equilibration cost from the per-sample cost. For energy estimation and more generally for observables commuting with the Hamiltonian, the required detailed-balanced measurements are implemented using Gaussian-filtered quantum phase estimation, with ancilla-qubit and Hamiltonian-simulation overhead logarithmic in precision parameters (Jiang et al., 13 Feb 2026).

"Single-Trajectory Gibbs Sampling for Non-Commuting Observables" extends this framework to arbitrary observables by two constructions (Chen et al., 23 Mar 2026). The first uses a three-Kraus measurement channel with

$x_{1:T}$05

plus a rejection Kraus operator $x_{1:T}$06, so that the resulting channel is CPTP and satisfies KMS detailed balance. In stationarity, if $x_{1:T}$07 when $x_{1:T}$08 or $x_{1:T}$09 fires and $x_{1:T}$10 when $x_{1:T}$11 fires, then

$x_{1:T}$12

is an unbiased estimator of $x_{1:T}$13 with $x_{1:T}$14. The paper proves that if the Gibbs sampler has spectral gap $x_{1:T}$15, then the integrated autocorrelation time of the resulting single-trajectory scheme is bounded by

$x_{1:T}$16

for some $x_{1:T}$17 (Chen et al., 23 Mar 2026).

The second construction sacrifices exact detailed balance but guarantees that each post-measurement state is a warm start in $x_{1:T}$18-divergence. It uses

$x_{1:T}$19

with estimator $x_{1:T}$20, and after each measurement applies the Gibbs sampler for

$x_{1:T}$21

steps to restore trace-distance proximity to $x_{1:T}$22. The total complexity becomes

$x_{1:T}$23

whereas the exact-detailed-balance version achieves the more direct replacement of $x_{1:T}$24 by $x_{1:T}$25 up to variance and precision factors (Chen et al., 23 Mar 2026). Taken together, these works establish that in the quantum setting the single-trajectory perspective is not merely an implementation detail but a distinct complexity regime based on autocorrelation-aware reuse of a stationary Gibbs trajectory.