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Non-asymptotic Estimates for Markov Transition Matrices via Spectral Gap Methods

De Huang, Xiangyuan Li

TL;DR

The paper provides non-asymptotic guarantees for estimating Markov transition matrices from finite samples by linking estimation error to spectral-type gaps. It introduces an induced length-2 path-space Markov chain $P_2$ to analyze the MLE error and develops a reversibility-preserving online SCE method for reversible chains, with associated non-asymptotic deviation bounds. Key results include entrywise tail bounds for the MLE error that depend on the iterated Poincaré gap $\eta_p(\bm{P})$, dimension-free Frobenius-norm bounds, and analogous matrix concentration bounds for SCE, along with detailed spectral-gap analyses of $P_2$ and $\tilde{P}_2$. Numerical experiments corroborate the theory, showing that the mean-squared error is largely dimension-free and scales with the sample size and spectral gap, highlighting the practical robustness of the proposed non-asymptotic framework.

Abstract

We establish non-asymptotic error bounds for the classical Maximal Likelihood Estimation of the transition matrix of a given Markov chain. Meanwhile, in the reversible case, we propose a new reversibility-preserving online Symmetric Counting Estimation of the transition matrix with non-asymptotic deviation bounds. Our analysis is based on a convergence study of certain Markov chains on the length-2 path spaces induced by the original Markov chain.

Non-asymptotic Estimates for Markov Transition Matrices via Spectral Gap Methods

TL;DR

The paper provides non-asymptotic guarantees for estimating Markov transition matrices from finite samples by linking estimation error to spectral-type gaps. It introduces an induced length-2 path-space Markov chain to analyze the MLE error and develops a reversibility-preserving online SCE method for reversible chains, with associated non-asymptotic deviation bounds. Key results include entrywise tail bounds for the MLE error that depend on the iterated Poincaré gap , dimension-free Frobenius-norm bounds, and analogous matrix concentration bounds for SCE, along with detailed spectral-gap analyses of and . Numerical experiments corroborate the theory, showing that the mean-squared error is largely dimension-free and scales with the sample size and spectral gap, highlighting the practical robustness of the proposed non-asymptotic framework.

Abstract

We establish non-asymptotic error bounds for the classical Maximal Likelihood Estimation of the transition matrix of a given Markov chain. Meanwhile, in the reversible case, we propose a new reversibility-preserving online Symmetric Counting Estimation of the transition matrix with non-asymptotic deviation bounds. Our analysis is based on a convergence study of certain Markov chains on the length-2 path spaces induced by the original Markov chain.
Paper Structure (20 sections, 12 theorems, 138 equations, 3 figures, 2 algorithms)

This paper contains 20 sections, 12 theorems, 138 equations, 3 figures, 2 algorithms.

Table of Contents

  1. introduction
  2. non-asymptotic estimates of markov matrices
  3. Fundamentals
  4. Notation
  5. Foundation of Markov chains
  6. Main results
  7. Generalized spectral gaps
  8. Related works
  9. Markov chains on path spaces
  10. Construction of $P_2$ and $\tilde{P}_2$
  11. Construction of $P_2$
  12. Construction of $\tilde{P}_2$
  13. Invariant distributions
  14. Unbiased estimation
  15. Spectral properties of $P_2$ and $\tilde{P}_2$
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $t>0$ and any $u,v\in\Omega$, where $C_1$, $C_2$ are absolute constants, $\nu$ is the initial distribution of the Markov sequence, and $\eta_p(\bm{P})$ is the IP gap of $\bm{P}$.

Figures (3)

  • Figure 5.1: Plots of $1/\text{MSE}$ as the function of $n$ for $\eta(\bm{P})=0.5,0.2,0.01$, respectively. We fix $N=10000$ and $d=50$.
  • Figure 5.2: Plots of $1/\text{MSE}$ as the function of $\eta(\bm{P})$ for $n=5000,20000,50000$, respectively. We fix $N=10000$ and $d=50$.
  • Figure 5.3: Plot of MSE as the function of $d$ for $N=5000$, $n=1000$. For each $d\in\left\{10,20,\cdots 200\right\}$, we take the average of the MSEs of 100 different $d\times d$ transition matrices with fixed spectral gap $\eta=0.1$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 18 more