We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up to a `coordinate transformation' consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. When the dynamics depends on an unknown parameter, we show that the latter can be estimated at the `standard' rate $n^{-1/2}$, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain `generator'. More generally, we show that the output is locally asymptotically normal, i.e. it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check we prove that a parameter related to the `coordinate transformation' unitaries, has zero quantum Fisher information.
We show that the posterior distribution of parameters in a hidden Markov model with parametric emission distributions and discrete and known state space is asymptotically normal. The main novelty of our proof is that it is based on a testing condition and the sequence of test functions is obtained using an optimal transportation inequality.
A Robbins-Monro algorithm for non-parametric estimation of NAR process with Markov-Switching: asymptotic normality
Aix-Marseille University（艾克斯-马赛大学）；Escuela de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela（委内瑞拉中央大学理学院数学系）；Departamento de Matemáticas, FACYT, Universidad de Carabobo（卡拉沃博大学FACYT数学系）
Generalized likelihoods are commonly used to obtain consistent estimators with attractive computational and robustness properties. Formally, any generalized likelihood can be used to define a generalized posterior distribution, but an arbitrarily defined "posterior" cannot be expected to appropriately quantify uncertainty in any meaningful sense. In this article, we provide sufficient conditions under which generalized posteriors exhibit concentration, asymptotic normality (Bernstein-von Mises), an asymptotically correct Laplace approximation, and asymptotically correct frequentist coverage. W…
This paper considers maximum likelihood (ML) estimation in a large class of models with hidden Markov regimes. We investigate consistency of the ML estimator and local asymptotic normality for the models under general conditions which allow for autoregressive dynamics in the observable process, Markov regime sequences with covariate-dependent transition matrices, and possible model misspecification. A Monte Carlo study examines the finite-sample properties of the ML estimator in correctly specified and misspecified models. An empirical application is also discussed.
University of Toronto（多伦多大学）；Vector Institute（向量学院）；University of California, Davis（加州大学戴维斯分校）
Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifica…
In the paper, we introduce the maximum entropy estimator based on 2-dimensional empirical distribution of the observation sequence of hidden Markov model , when the sample size is big: in that case computing the maximum likelihood estimator is too consuming in time by the classical Baum-Welch EM algorithm. We prove the consistency and the asymptotic normality of the maximum entropy estimator in a quite general framework, where the asymptotic covariance matrix is explicitly estimated in terms of the 2-dimensional Fisher information. To complement it, the 2-dimensional relative entropy is skillf…
Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Daniels School of Business, Purdue University（普渡大学丹尼尔斯商学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）
High-dimensional Inference and FDR Control for Simulated Markov Random Fields
University of California San Diego（加利福尼亚大学圣迭戈分校）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；University of Alberta（阿尔伯塔大学）；Beihang University（北京航空航天大学）；Zhuhai UM Science & Technology Research Institute（珠海澳大科技研究院）
We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimizati…
We estimate a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the transition intensity matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The new mixture is estimated from its continuously observed realizations using the EM algorithm, which provides the maximum likelihood (ML) estimates of the mixture's parameters. We derive the asymptotic properties of the ML estimators. To obtain estimated standard errors o…
Equivalence of the measures of non-Markovianty for open two-level systems
为了刻画开放系统中量子时间演化偏离马尔可夫过程的程度，人们提出了不同的度量。我们证明了Breuer，Laine和Piilo [Phys. Rev. Lett. 103，210401（2009）]和Rivas，Huelga和Plenio [Phys. Rev. Lett. 105，050403（2010）]具有完全相同的非马尔可夫时间演化间隔，因此当它们应用于通过Jaynes-Cummings模型或退相模型耦合到环境的开放两能级系统时，它们实际上彼此等价。这种等价性意味着这三种测度以不同的方式捕捉到了量子演化过程的非马尔可夫性的本质特征。我们还表明，在第一个措施的定义中的最大化，实际上可以删除所考虑的模型，而不影响检测非马尔可夫性的措施的灵敏度。
