David J. Aldous, Antar Bandyopadhyay
Use each of n exact samples as the initial state for a MCMC sampler run for m steps. We give confidence intervals for accuracy of estimators which are always valid and which, in certain settings, are almost as good as the intervals one would obtain if the (unknown) mixing time of the chain were known.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp.
Slow diffusion by Markov random flights
本文提出基于小速率齐次泊松过程驱动的小恒定速度随机飞行理论的欧几里得空间慢扩散过程概念，给出了长时间区间上导致平稳分布的条件，并展示了低维空间中的平稳分布。
We study a general $k$ dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index $α\in (0,1)$. When the arrival rate is sped up by a factor $n^γ$, the transition probabilities of the underlying Markov chain are divided by $n^γ$ and the service times are divided by $n$, we identify two regimes (''fast arrivals'', when $γ>α$, and ''equilibrium'', when $γ=α$) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third ''slow arrivals'' regime, $γ<α$, we sh…
This paper is concerned with the development of rigorous approximations to various expectations associated with Markov chains and processes having non-stationary transition probabilities. Such non-stationary models arise naturally in contexts in which time-of-day effects or seasonality effects need to be incorporated. Our approximations are valid asymptotically in regimes in which the transition probabilities change slowly over time. Specifically, we develop approximations for the expected infinite horizon discounted reward, the expected reward to the hitting time of a set, the expected reward…
Exponential stability of the nonlinear filtering equation is revisited, when the signal is a finite state Markov chain. An asymptotic upper bound for the filtering error due to incorrect initial condition is derived in the case of slowly switching signal.
Consider a sequence $(η^N(t) :t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(η,ξ)$ the jump rates of the Markov chain $η^N_t$, and assume that for any pair of bonds $(η,ξ)$, $(η',ξ')$ $\arctan \{R_N(η,ξ)/R_N(η',ξ')\}$ converges as $N\uparrow\infty$. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence of increasing time-scales $θ^1_N, \dots, θ^{\mf p}_N$, $θ^j_N \ll θ^{j+1}_N$, and of coarsening partitions $\{\ms E^j_1, \dots, \ms E…
In this paper, we consider a general class of two-time-scale Markov chains whose transition rate matrix depends on a parameter $λ>0$. We assume that some transition rates of the Markov chain will tend to infinity as $λ\rightarrow\infty$. We divide the state space of the Markov chain $X$ into a fast state space and a slow state space and define a reduced chain $Y$ on the slow state space. Our main result is that the distribution of the original chain $X$ will converge in total variation distance to that of the reduced chain $Y$ uniformly in time $t$ as $λ\rightarrow\infty$.
Boundary-induced slow mixing for Markov chains and its application to stochastic reaction networks
本文研究了非负象限上马尔可夫链的边界诱导慢混合现象，建立了混合时间幂律下界的通用准则，并揭示了化学反应网络结构对慢混合的影响。
The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow mot…
In [20], the authors addressed the question of the averaging of a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimension. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuation of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated Langevin approximation is considered. The motivation of this work is a stochastic Hodgkin-Huxley model which describes the propagation of an action potential along the nerve fiber. We study this PDMP in detail and provide more general results fo…
Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly
该研究运用CAT(0)立方复形理论证明带状区域单调路径上两类马尔可夫链混合缓慢，还给出了单调路径计数公式并推广了相关指数增长常数结果。
Flowshop Machine Scheduling: Markov Modeling, Optimal Schedules and Heuristics
本文提出一种新颖的马尔可夫链模型，用于分析存在等效故障的 N/M/P/F 置换流水车间调度问题，推导了完工时间的闭式表达式，并提出了生成最优或近最优调度的启发式算法。
Moderate Deviation Principles for Stochastic Differential Equations in Fast-Varying Markovian Environment
本文研究了在快速变化的马尔可夫环境中，双时间尺度随机系统中等偏差原理的证明方法，结合弱收敛方法与泊松方程，解决了慢过程和快过程的耦合问题。
We prove pathwise large deviation principles of slow variables in slow-fast systems in the limit of time-scale separation tending to infinity. In the limit regime we consider, the convergence of the slow variable to its deterministic limit and the convergence of the fast variable to equilibrium are competing at the same scale. The large deviation principle is proven by relating the large deviation problem to solutions of Hamilton-Jacobi-Bellman equations, for which well-posedness was established in the companion paper [arXiv:1912.06579]. We cast the rate functions in action-integral form and i…
On the study of slow-fast dynamics, when the fast process has multiple invariant measures
本文研究快过程具有多个不变测度的慢-快随机系统平均化问题，在解耦与完全耦合及有限状态连续时间马尔可夫快过程设定下，证明极限动力学为随机 ODE。
Consider a sequence of i.i.d. random Lipschitz functions $\{Ψ_n\}_{n \geq 0}$. Using this sequence we can define a Markov chain via the recursive formula $R_{n+1} = Ψ_{n+1}(R_n)$. It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when $Ψ_0(t) \approx A_0t+B_0$. We will show that under subexponential assumptions on the random variable $\log^+(A_0\vee B_0)$ the tail asymptotic in question can be described using the integrated tail function of $\log^+(A_0\…
In this paper, we consider the control problem with the Average-Value-at-Risk (AVaR) criteria of the possibly unbounded $L^{1}$-costs in infinite horizon on a Markov Decision Process (MDP). With a suitable state aggregation and by choosing a priori a global variable $s$ heuristically, we show that there exist optimal policies for the infinite horizon problem. To our knowledge, this is the first work of deriving dynamic programming equations with $L^1$-unbounded costs via AVaR-operator.
In this paper we consider the convergence of the conditional entropy to the entropy rate for Markov chains. Convergence of certain statistics of long range dependent processes, such as the sample mean, is slow. It has been shown in Carpio and Daley \cite{carpio2007long} that the convergence of the $n$-step transition probabilities to the stationary distribution is slow, without quantifying the convergence rate. We prove that the slow convergence also applies to convergence to an information-theoretic measure, the entropy rate, by showing that the convergence rate is equivalent to the convergen…
It is known that random walk Metropolis algorithms with heavy-tailed target densities can model atypical (slow) growth of maxima, which in general is exhibited by processes with the extremal index zero. The asymptotics of maxima of such sequences can be analyzed in terms of continuous phantom distribution functions. We show that in a large class of positive Harris recurrent Markov chains (containing the above Metropolis chains) a phantom distribution function can be recovered by starting "at the point" rather than from the stationary distribution.
We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process in natural scale can be approximated with such Markov chains arbitrarily well. The functional limit theorem applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. Our results allow to practically approximate such processes with irregular behavior; we illustrate thi…
A curiously slowly mixing Markov chain
该研究提出了一种超立方体上的Burnside过程，展示了其在不同混合度量下的不同混合速率，并通过显式对角化方法解释了$\ell^2$混合结果。
We consider Piecewise Deterministic Markov Processes (PDMPs) with a finite set of discrete states. In the regime of fast jumps between discrete states, we prove a law of large number and a large deviation principle. In the regime of fast and slow jumps, we analyze a coarse-grained process associated to the original one and prove its convergence to a new PDMP with effective force fields and jump rates. In all the above cases, the continuous variables evolve slowly according to ODEs. Finally, we discuss some applications related to the mechanochemical cycle of macromolecules, including strained-…
We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.
The Arnoldi Aggregation for Approximate Transient Distributions of Markov Chains
本文提出基于Arnoldi迭代的非划分式马尔可夫链瞬态分布聚合方法，证明了其精确性与规模最小性，设计了精度达标后停止扩展的启发式规则，实验验证其效果优于基于lumping的状态约简方案。
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
研究平面图$\alpha$-定向上的上下马尔可夫链的混合时间，证明了在特定四边形图和三角形剖分上存在慢混合，而在最大度至多4的四边形图上快速混合。
We consider sampling and enumeration problems for Markov equivalence classes. We create and analyze a Markov chain for uniform random sampling on the DAGs inside a Markov equivalence class. Though the worst case is exponentially slow mixing, we find a condition on the Markov equivalence class for polynomial time mixing. We also investigate the ratio of Markov equivalence classes to DAGs and a Markov chain of He, Jia, and Yu for random sampling of sparse Markov equivalence classes.
We prove finite sample complexities for sequential Monte Carlo (SMC) algorithms which require only local mixing times of the associated Markov kernels. Our bounds are particularly useful when the target distribution is multimodal and global mixing of the Markov kernel is slow; in such cases our approach establishes the benefits of SMC over the corresponding Markov chain Monte Carlo (MCMC) estimator. The lack of global mixing is addressed by sequentially controlling the bias introduced by SMC resampling procedures. We apply these results to obtain complexity bounds for approximating expectation…
Rates of forgetting for the sequentially Markov coalescent
本文研究了序列马尔可夫凝聚过程在配对设定下的遗忘速率，证明了嵌入跳链的几何平稳性，并展示了连续过程在遗传距离下的平稳距离衰减规律，同时通过新颖的时间变换论证得出了SMC'过程的类似结果。
Locally Stationary Distributions: A Framework for Analyzing Slow-Mixing Markov Chains
本文提出局部平稳分布框架，用于分析慢混合马尔可夫链，并展示其在 hardcore 模型独立集、带秩一尖峰的 Ising 模型及双社区随机块模型中实现常数相关性的算法应用。
The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require…
Convergence Bounds for Monte Carlo Markov Chains
本文为新手介绍马尔可夫链蒙特卡洛的收敛分析，重点通过耦合方法和基于电导与等周不等式的L2理论构建链分布与平稳分布间距离的界，并讨论识别缓慢收敛的策略。
Schur--Weyl duality for diagonalizing a Markov chain on the hypercube
本文利用Schur-Weyl对偶等工具，为超立方体上特定马尔可夫链提供显式正交基，并确定其收敛到平稳分布的精确速率。
We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of proof, using techniques from empirical process theory, does not rely on the explicit expression for the maximum likelihood estimator in the Markov case and could therefore be applicable in other setting…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。