Giovanni Pistone, Maria Piera Rogantin
For a Markov chain both the detailed balance condition and the cycle Kolmogorov condition are algebraic binomials. This remark suggests to study reversible Markov chains with the tool of Algebraic Statistics, such as toric statistical models. One of the results of this study in an algebraic parameterization of reversible Markov transitions and their invariant probability.
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I show how any reversible Markov chain on a finite state space that is irreducible, and hence suitable for estimating expectations with respect to its invariant distribution, can be used to construct a non-reversible Markov chain on a related state space that can also be used to estimate these expectations, with asymptotic variance at least as small as that using the reversible chain (typically smaller). The non-reversible chain achieves this improvement by avoiding (to the extent possible) transitions that backtrack to the state from which the chain just came. The proof that this modification…
Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key rôle in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterize and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are scarce. We show that for a type of nonreversible Monte Carlo Markov chains and processes, of current or renewed interest in th…
We define a conjugate prior for the reversible Markov chain of order $r$. The prior arises from a partially exchangeable reinforced random walk, in the same way that the Beta distribution arises from the exchangeable Polyá urn. An extension to variable-order Markov chains is also derived. We show the utility of this prior in testing the order and estimating the parameters of a reversible Markov model.
In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.
Let $S_N$ be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein--Hoeffding inequality is derived for the probability of large deviations of $S_N$ and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.
Graph Disjointness with Applications to Reversible Markov Chains
本文研究图与马尔可夫链的不相交性，通过图连接理论分析其结构差异，提出弱不相交和强不相交的定义，并揭示其与图因子的关系。
We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions…
Improved Estimation of Relaxation Time in Non-reversible Markov Chains
本文确定了常数乘法误差下遍历马尔可夫链伪谱隙估计的极小化样本复杂度，解决了相关开放问题，并通过完全自适应数据改进了经验过程，收窄了置信区间并降低了计算复杂度。
Reversible Markov chains: variational representations and ordering
本文教学性地阐述比较可逆马尔可夫链效率的三种变分表示，包括谱间隙、渐近方差和电导，并证明非零电导等价于非零右谱间隙。
We introduce a natural conjugate prior for the transition matrix of a reversible Markov chain. This allows estimation and testing. The prior arises from random walk with reinforcement in the same way the Dirichlet prior arises from Pólya's urn. We give closed form normalizing constants, a simple method of simulation from the posterior and a characterization along the lines of W. E. Johnson's characterization of the Dirichlet prior.
Let $(X_t)$ be a discrete time Markov chain on a general state space. It is well-known that if $(X_t)$ is aperiodic and satisfies a drift and minorization condition, then it converges to its stationary distribution $π$ at an exponential rate. We consider the problem of computing upper bounds for the distance from stationarity in terms of the drift and minorization data. Baxendale showed that these bounds improve significantly if one assumes that $(X_t)$ is reversible with nonnegative eigenvalues (i.e. its transition kernel is a self-adjoint operator on $L^2(π)$ with spectrum contained in $[0,1…
The spectral gap $γ$ of an ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $t$ may be observed. Hsu, Kontorovich, and Szepesvari (2015) considered the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. They showed that, if $t = \tilde{O}\bigl(\frac{1}{γ^3 π_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. They also …
We introduce a three-parameter random walk with reinforcement, called the $(θ,α,β)$ scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter $β$ smoothly tunes the $(θ,α,β)$ scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters $α$ and $θ$ modulate how many states are typically visited. Resorting to de Finetti's theorem for Markov chains, we use the $(θ,α,β)$ scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in…
We consider two approaches to study non-reversible Markov processes, namely the Hypocoercivity Theory (HT) and GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling); the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker-Plan…
We describe ergodic properties of some Metropolis-Hastings (MH) algorithms for heavy-tailed target distributions. The analysis usually falls into sub-geometric ergodicity framework but we prove that the mixed preconditioned Crank-Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that the MpCN algorithm becomes a random-walk Metropolis algorithm under suitable transformation.
We study the problem of hypothesis testing between two discrete distributions, where we only have access to samples after the action of a known reversible Markov chain, playing the role of noise. We derive instance-dependent minimax rates for the sample complexity of this problem, and show how its dependence in time is related to the spectral properties of the Markov chain. We show that there exists a wide statistical window, in terms of sample complexity for hypothesis testing between different pairs of initial distributions. We illustrate these results in several concrete examples.
We study $k$-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer $k=1,2,\ldots$. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for $k>1$, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable $k$-divisible partitions that are consistent under random deletio…
Geometric ergodicity of trans-dimensional Markov chain Monte Carlo algorithms
本文研究了模型总数有限时跨维度MCMC算法的收敛性，证明在温和条件下，若模型内移动的马尔可夫链几何遍历，则可逆及部分非可逆跨维度链可保证几何收敛，并将该理论应用于三种贝叶斯模型。
Relaxation times of non-reversible Markov processes
研究非可逆马尔可夫过程的弛豫时间，基于生成器奇异值间隙开发系统方法，能量化弛豫时间，给出精确上下界，还可推导奇异值间隙下界，应用于证明猜想及给出相关过程弛豫时间的界。
Efficient shape-constrained inference for the autocovariance sequence from a reversible Markov chain
针对可逆马尔可夫链自协方差序列估计问题，提出基于矩序列形状约束的新型估计器，证明其强一致性并通过实验验证优于现有主流方法。
Spectral gap bounds for reversible hybrid Gibbs chains
本文针对可逆混合Gibbs采样器收敛性质研究不足的问题，提出新方法建立其收敛速率界，证明混合Gibbs链绝对谱间隙可由精确链及近似用马尔可夫链的谱间隙界定，并拓展至混合数据增强算法，应用于三类采样器实例。
Rebalancing Markov jump processes for non-reversible continuous-time sampling
针对马尔可夫链蒙特卡罗方法因细致平衡导致类扩散探索需精细调参的问题，提出无需细致平衡的非可逆连续时间采样器构建机制，通过转化满足斜细致平衡的跳跃过程并利用无界平衡函数动态选跃迁，经实验验证其参数调整稳健性更优。
We consider the problem of identity testing of Markov chain transition matrices based on a single trajectory of observations under the distance notion introduced by Daskalakis et al. [2018a] and further analyzed by Cherapanamjeri and Bartlett [2019]. Both works made the restrictive assumption that the Markov chains under consideration are symmetric. In this work we relax the symmetry assumption and show that it is possible to perform identity testing under the much weaker assumption of reversibility, provided that the stationary distributions of the reference and of the unknown Markov chains a…
We consider dependence coefficients for stationary Markov chains. We emphasize on some equivalencies for reversible Markov chains. We improve some known results and provide a necessary condition for Markov chains based on Archimedean copulas to be exponential $ρ$-mixing. We analyze the example of the Mardia and Frechet copula families using small sets.
Markov chain Monte Carlo methods have become standard tools in statistics to sample from complex probability measures. Many available techniques rely on discrete-time reversible Markov chains whose transition kernels build up over the Metropolis-Hastings algorithm. We explore and propose several original extensions of an alternative approach introduced recently in Peters and de With (2012) where the target distribution of interest is explored using a continuous-time Markov process. In the Metropolis-Hastings algorithm, a trial move to a region of lower target density, equivalently "higher ener…
We address the problem of estimating the mixing time $t_{\mathsf{mix}}$ of an arbitrary ergodic finite-state Markov chain from a single trajectory of length $m$. The reversible case was addressed by Hsu et al. [2019], who left the general case as an open problem. In the reversible case, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl's inequality allows for a dimension-free perturbation analysis of the empirical eigenvalues. As Hsu et al. point out, in the absence of reversibility (which induces asymmetric pair probabilities matrices), the exi…
The spectral gap $γ$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. We show that if $n = \tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. When $π$ is uniform on $d$ states, this matc…
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