Decoupling for Markov Chains

TL;DR

<3-5 sentences high-level summary of the paper>: The work introduces a tangent–decoupled framework for Markov chains generated via i.i.d. auxiliary randomness, enabling a companion process with simplified temporal dependence while preserving the target marginal μ. It proves that this decoupled sequence provides a consistent estimator of μ(f) and establishes a sharp, uniform finite–sample variance bound σ_f^2 ≤ 2~σ_f^2 that holds without reversibility or mixing assumptions. The authors leverage a sharp L^2 tangent decoupling inequality to show the bound is tight (as demonstrated in AR(1) examples) and derive practical uncertainty quantification tools, including confidence intervals that are conservatively valid for a wide class of MCMC algorithms. Through AR(1) case studies and MCMC applications, the paper demonstrates how tangent decoupling can yield reliable Monte Carlo error assessments and suggests broader uses in concentration inequalities and algorithm design.

Abstract

Consider a Markov chain $(X_i)_{i\ge0}$ with invariant measure $μ$ that admits the representation $X_{i+1}=Φ(X_i,U_i)$, where $(U_i)_{i\ge0}$ are i.i.d. random variables and $Φ$ is a measurable map. We introduce a tangent-decoupled process $(\widetilde X_i)_{i\ge0}$ obtained by replacing $(U_i)$ with an independent copy. Conditional on the realized backbone $(X_i)$, the sequence $(f(\widetilde X_i))$ is independent. Although $(\widetilde X_i)$ is not Markovian, under the same ergodicity assumptions that ensure a law of large numbers for $(X_i)$, the empirical averages $n^{-1}\sum_{i=1}^n f(\widetilde X_i)$ converge almost surely to $μ(f)$. In addition, for every $f\in L^2(μ)$ and every $N\ge1$, $$ \operatorname{Var}\!\Bigl(\sum_{i=1}^N f(X_i)\Bigr) \;\le\; 2\,\operatorname{Var}\!\Bigl(\sum_{i=1}^N f(\widetilde X_i)\Bigr), $$ and therefore $σ_f^2 \le 2\,\widetildeσ_f^{\,2}$ for the corresponding time-average variance constants. The inequality requires neither reversibility nor mixing assumptions. Its proof identifies the two sequences as tangent in the sense of decoupling theory and applies the sharp $L^2$ tangent decoupling inequality of de la Peña, Yao, and Alemayehu (2025).

Figures (4)

• Figure 1: Propagation of auxiliary randomness in the original chain (panel (a)) versus the tangent--decoupled chain (panel (b)). In the original chain, $U_0$ influences all downstream states. In the tangent--decoupled chain, $\widetilde{U}_0$ affects only one transition.
• Figure 2: Consistency of the tangent–decoupled estimator in the AR(1) example. Each panel shows the running mean $\frac{1}{n}\sum_{i=1}^n f(X_i)$ (solid gray) and its tangent–decoupled analogue $\frac{1}{n}\sum_{i=1}^n f(\widetilde{X}_i)$ (solid red), computed along a single long trajectory of length $N=10^5$. The black line marks the true expectation $\mu(f)$. Left: $f(x)=x$. Middle: $f(x)=x^2$. Right: $f(x)=\mathbf{1}_{\{x>c\}}$ with $c=-0.5$. In all cases the tangent–decoupled running mean converges to the correct limit.
• Figure 3: Comparison of the empirical time–average variance estimate $N\,\operatorname{Var}(\bar{f}_N)$ for the AR(1) chain (solid black), its tangent–decoupled analogue $2N\,\operatorname{Var}(\bar{f}_N^{\,\widetilde{X}})$ (dashed black), and the corresponding theoretical TAVC $\sigma_f^{\,2}$ (red dotted), for the observables $f(x)=x$ (left), $f(x)=x^2$ (middle, shown on a logarithmic scale), and the step function $f(x)=\mathbf{1}_{\{x>c\}}$ with $c=-0.5$ (right). In all cases the tangent–decoupled sequence exhibits weaker temporal dependence, and the empirical behavior clearly reflects Theorem \ref{['thm:variance']}.
• Figure 4: Estimated long--run variance $\widehat{\sigma}^2(m)$ as a function of the truncation parameter $m$ for NUTS applied to a $d=100$ Toeplitz Gaussian target with correlation parameter $\rho=0.9$. Results are shown for the three test functions $f_1(\theta)=d^{-1}\sum_i \theta_i$, $f_2(\theta)=d^{-1}\sum_i \theta_i^2$, and $f_3(\theta)=\bigl(d^{-1/2}\sum_i \theta_i\bigr)^2$, comparing the standard backbone chain (gray) and the tangent--decoupled sequence (red). Variance estimates are computed using Geyer’s initial positive sequence estimator based on unbiased autocovariances. The empirical means of both the standard and decoupled sequences are close to their known true values, with the decoupled estimator exhibiting smaller relative error across all three test functions.

Theorems & Definitions (25)

• Definition 1.1: Tangent-decoupled sequence
• Theorem 1.2: Consistency of the tangent--decoupled estimator
• Theorem 1.3: Finite--sample and asymptotic variance comparison
• Lemma 2.1: Conditional independence
• proof
• Lemma 2.2: Stationary marginals under tangent decoupling
• proof
• Theorem 2.3: Consistency of the tangent--decoupled estimator
• proof
• Remark 2.4: Effect of averaging multiple tangent draws