A family $\{Q_β\}_{β\geq 0}$ of Markov chains is said to exhibit $\textit{metastable mixing}$ with $\textit{modes}$ $S_β^{(1)},\ldots,S_β^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min(Φ_β(S_β^{(1)}), \ldots, Φ_β(S_β^{(k)}))$ of its modes. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis-Hastings chain targeting a mixture distribution. Our work differs from existing work on metastability in that, for the class of examples we are interested in, it gives an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp) while at the same time giving technical conditions that are easier to verify for many statistical examples. Our bounds from this paper are used in a companion paper to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.
We develop a theory of weak Poincaré inequalities to characterize convergence rates of ergodic Markov chains. Motivated by the application of Markov chains in the context of algorithms, we develop a relevant set of tools which enable the practical study of convergence rates in the setting of Markov chain Monte Carlo methods, but also well beyond.
We introduce an ensemble Markov chain Monte Carlo approach to sampling from a probability density with known likelihood. This method upgrades an underlying Markov chain by allowing an ensemble of such chains to interact via a process in which one chain's state is cloned as another's is deleted. This effective teleportation of states can overcome issues of metastability in the underlying chain, as the scheme enjoys rapid mixing once the modes of the target density have been populated. We derive a mean-field limit for the evolution of the ensemble. We analyze the global and local convergence of …
Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$. The weights $θ^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{(p)}$ identify the me…
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. In Part I of this work, by applying a…
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. Applying a Markov mapping theorem, we…
A Graph-Algorithmic Approach for the Study of Metastability in Markov Chains
具有指数小转移率的大型连续时间马尔可夫链出现在物理、化学和生物学中的复杂系统建模中。我们提出了一个建设性的图形算法的方法来确定一个给定的马尔可夫链的定性行为的变化的关键时间尺度的序列，并给出了一个有效的描述他们每个人的动态。这种方法是有效的时间可逆和时间不可逆的马尔可夫过程，或不对称。该方法的核心是两个图算法，算法1和算法2，用于获得临界时标的序列和典型的过渡图或T-图的层次结构，分别指示系统{无对称性和有对称性}中最可能的过渡。{临界}时标序列包括特征值的{实部}倒数的子序列。在一定的假设下，我们证明尖锐的渐近估计特征值（包括前因子），并显示如何可以提取它们从算法1的输出。我们讨论了算法1和算法2之间的关系，并解释了如果算法1而不是算法2应用于对称性的情况下，如何解释算法1的输出。最后，我们分析了一个由R. D. Astumian的动力学模型，一种分子马达，通过算法2。
We consider a random walk with catastrophes which was introduced to model population biology. It is known that this Markov chain gets eventually absorbed at $0$ for all parameter values. Recently, it has been shown that this chain exhibits a metastable behavior in the sense that it can persist for a very long time before getting absorbed. In this paper we study this metastable phase by making the parameters converge to extreme values. We obtain four different limits that we believe shed light on the metastable phase.
Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz-Thouless phase transition in the two dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions; but rather than concentrating on the equilibrium measure we try to understand its dynamics. The equivalent of the Kosterlitz-Thouless transition in this one dimensional case happens when the interaction strength scales like the size of the system $N$, yet we show that a sharp transition for…
We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these pha…
This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof …
The asymptotic variance is an important criterion to evaluate the performance of Markov chains, especially for the central limit theorems. We give the variational formulas for the asymptotic variance of discrete-time (non-reversible) Markov chains on general state space. The variational formulas provide many applications, extending the classical Peskun's comparison theorem to non-reversible Markov chains, and obtaining several comparison theorems between Markov chains with various perturbations.
We study inhomogeneous continuous-time weakly ergodic Markov chains with a finite state space. We introduce the notion of a Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of convergence for such class of Markov chains.
In this paper, we present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility.
Extensions of Kemeny's constant, as derived for irreducible finite Markov chains in discrete time, to Markov renewal processes and Markov chains in continuous time are discussed. Three alternative Kemeny's functions and their variants are considered. Typically, they lead to a constant if and only if the mean holding times between the states in the Markov renewal process are constant. However one particular variant leads to a constant, analogous to the discrete time Markov chain result. Specifically, if the state space is finite, the weighted sum of the mean first passage times (omitting the me…
We consider the problem of characterising expected hitting times and hitting probabilities for imprecise Markov chains. To this end, we consider three distinct ways in which imprecise Markov chains have been defined in the literature: as sets of homogeneous Markov chains, as sets of more general stochastic processes, and as game-theoretic probability models. Our first contribution is that all these different types of imprecise Markov chains have the same lower and upper expected hitting times, and similarly the hitting probabilities are the same for these three types. Moreover, we provide a ch…
We consider periodic Markov chains with absorption. Applying to iterates of this periodic Markov chain criteria for the exponential convergence of conditional distributions of aperiodic absorbed Markov chains, we obtain exponential estimates for the periodic asymptotic behavior of the semigroup of the Markov chain. This implies in particular the exponential convergence in total variation of the conditional distribution of the Markov chain given non-absorption to a periodic sequence of limit measures and we characterize the cases where this sequence is constant, which corresponds to the cases w…
We study an approximation method of stationary characters of a two-dimensional Markov chain via the Stein method. For this purpose, innovative methods are developed to estimate the moments of the Markov chain, as well as the solution to the Poisson equation with a partial differential operator.
In this paper, we quantify some known approximation to the Curie-Weiss model via applying the Stein method to the Markov chain whose stationary distribution coincides with Curie-Weiss model.
In many dynamical systems in nature, the law of the dynamics changes along with the temporal evolution of the system. These changes are often associated with the occurrence of certain events. The timing of occurrence of these events depends, in turn, on the trajectory of the dynamical system itself, making the dynamics of the system and the timing of a change in the dynamics strongly coupled. Naturally, trajectories that take longer to satisfy the event will last longer. Therefore, we expect to observe more frequently the dominant dynamics, the ones that take longer to change in the long run. …
