Markov bases of binary graph models

Effects of statistical dependence on multiple testing under a hidden Markov model

The performance of multiple hypothesis testing is known to be affected by the statistical dependence among random variables involved. The mechanisms responsible for this, however, are not well understood. We study the effects of the dependence structure of a finite state hidden Markov model (HMM) on the likelihood ratios critical for optimal multiple testing on the hidden states. Various convergence results are obtained for the likelihood ratios as the observations of the HMM form an increasing long chain. Analytic expansions of the first and second order derivatives are obtained for the case …

Central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistence and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations…

Maximum likelihood fitting of acyclic directed mixed graphs to binary data

Acyclic directed mixed graphs, also known as semi-Markov models represent the conditional independence structure induced on an observed margin by a DAG model with latent variables. In this paper we present the first method for fitting these models to binary data using maximum likelihood estimation.

Estimation and Applications of Quantile Regression for Binary Longitudinal Data

This paper develops a framework for quantile regression in binary longitudinal data settings. A novel Markov chain Monte Carlo (MCMC) method is designed to fit the model and its computational efficiency is demonstrated in a simulation study. The proposed approach is flexible in that it can account for common and individual-specific parameters, as well as multivariate heterogeneity associated with several covariates. The methodology is applied to study female labor force participation and home ownership in the United States. The results offer new insights at the various quantiles, which are of …

Assessment of school performance through a multilevel latent Markov Rasch model

An extension of the latent Markov Rasch model is described for the analysis of binary longitudinal data with covariates when subjects are collected in clusters, e.g. students clustered in classes. For each subject, the latent process is used to represent the characteristic of interest (e.g. ability) conditional on the effect of the cluster to which he/she belongs. The latter effect is modeled by a discrete latent variable associated with each cluster. For the maximum likelihood estimation of the model parameters we outline an EM algorithm. We show how the proposed model may be used for assessi…

Markov bases of binary graph models of K_4-minor free graphs

Markov width of a graph is a graph invariant defined as the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We show that a graph has Markov width at most four if and only if it contains no $K_4$ as a minor, answering a question of Develin and Sullivant. We also present a lower bound of order $Ω(n^{2-\varepsilon})$ on the Markov width of $K_n$.

马尔可夫链的信息散度及其应用

Information divergences of Markov chains and their applications

本文定义有限马尔可夫链转移矩阵上的信息散度，推导其 Pinsker 不等式等性质，并用于二元假设检验、Cesàro 混合时间与遍历性系数分析。

描述在二元碰撞机制下演化的$N$粒子系统的某些量子 Markov 半群的谱

Spectrum for some Quantum Markov semigroups describing $N$-particle systems evolving under a binary collision mechanism

本文研究二元碰撞$N$粒子系统对应的量子 Markov 半群，揭示其生成元谱与图 Laplacian 谱的联系，并精确求得 Kac 模型量子类比的谱隙。

L-cumulants, L-cumulant embeddings and algebraic statistics

Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization keeps all the desired properties of the classical cumulants like semi-invariance and vanishing for independent blocks of random variables. These properties make L-cumulants useful for the algebraic analysis of statistical models. We illustrate this for general Markov models and hidden Markov processes in the case when the hidden process is binary. The main motivation of this work is to understand cumulant-like coordinates in algebraic statistics and to give…

对超立方体上马尔可夫链对角化中的Schur-Weyl对偶

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。