Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations

图模型的马尔可夫链蒙特卡洛扰动分析

Perturbation Analysis of Markov Chain Monte Carlo for Graphical Models

本文研究了图模型中马尔可夫链蒙特卡洛的扰动分析，证明对于图模型相关的目标分布，允许比一般情况大至收敛速率平方根的扰动，且近似MCMC算法中更大的扰动仍能保证较小误差。

Graphical Markov models for infinitely many variables

Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local and global Markov properties demonstrated by Pearl and Paz (1985). Modern day applications often necessitate working with eithe…

Time-like graphical models

We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. Burdzy and Pal in their paper proposed a continuous version of graphical models -- processes indexed by time-like graphs. We construct a more general family of continuous processes on a wider family of graphs, find new properties and solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We present new results showing connections to the stochastic heat equation, Markov random…

A polyhedral Markov field - pushing the Arak-Surgailis construction into three dimensions

The purpose of the paper is to construct a polyhedral Markov field in ${\mathbb R}^3$ in analogy with the planar construction of the original Arak (1982) polygonal Markov field. We provide a dynamic construction of the process in terms of evolution of two-dimensional multi-edge systems tracing polyhedral boundaries of the field in three-dimensional time-space. We also give a general algorithm for simulating Gibbsian modifications of the constructed polyhedral field.

Two generalizations of Markov blankets

In a probabilistic graphical model on a set of variables $V$, the Markov blanket of a random vector $B$ is the minimal set of variables conditioned to which $B$ is independent from the remaining of the variables $V \backslash B$. We generalize Markov blankets to study how a set $C$ of variables of interest depends on~$B$. Doing that, we must choose if we authorize vertices of $C$ or vertices of $V \backslash C$ in the blanket. We therefore introduce two generalizations. The Markov blanket of $B$ in $C$ is the minimal subset of $C$ conditionally to which $B$ and $C$ are independent. It is natur…

Systemic Infinitesimal Over-dispersion on Graphical Dynamic Models

Stochastic models for collections of interacting populations have crucial roles in scientific fields such as epidemiology and ecology, yet the standard approach to extending an ordinary differential equation model to a Markov chain does not have sufficient flexibility in the mean-variance relationship to match data. To handle that, over-dispersed Markov chains have previously been constructed using gamma white noise on the rates. We develop new approaches using Dirichlet noise to construct collections of independent or dependent noise processes. This permits the modeling of high-frequency vari…

Uncertainty quantification for Markov Random Fields

We present an information-based uncertainty quantification method for general Markov Random Fields. Markov Random Fields (MRF) are structured, probabilistic graphical models over undirected graphs, and provide a fundamental unifying modeling tool for statistical mechanics, probabilistic machine learning, and artificial intelligence. Typically MRFs are complex and high-dimensional with nodes and edges (connections) built in a modular fashion from simpler, low-dimensional probabilistic models and their local connections; in turn, this modularity allows to incorporate available data to MRFs and e…

Non-existence of Markovian time dynamics for graphical models of correlated default

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be re-rewritten as a standard Ising model on the graph by identifying the set of d…

Extremes of Markov random fields on block graphs: max-stable limits and structured Hüsler-Reiss distributions

We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. The latter limit relation implies that the random field is multivariate regularly varying and it determines the max-stable …

Gaussian Covariance faithful Markov Trees

A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge between any pair of variables implies marginal independence between these two variables. Covariance graph models have recently received much attention in the literature and constitute a sub-family of graphical models. Though they are conceptually simple to understand, they are considerably more difficult to analyze. Under some suitable assumption on the proba…