Q-Markov random probability measures and their posterior distributions

Characterization of stationary probability measures for Variable Length Markov Chains

By introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure.

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d. environment to the Markovian one the classical classification of the branching processes into critical and strongly, intermediate and weakly subcritical states. In all these cases, we study the asymptotic behaviour of the probability that $Z_n>0$ as $n\to+\infty$.

Quantitative approximations of evolving probability measures and sequential Markov Chain Monte Carlo methods

We study approximations of evolving probability measures by an interacting particle system. The particle system dynamics is a combination of independent Markov chain moves and importance sampling/resampling steps. Under global regularity conditions, we derive non-asymptotic error bounds for the particle system approximation. In a few simple examples, including high dimensional product measures, bounds with explicit constants of feasible size are obtained. Our main motivation are applications to sequential MCMC methods for Monte Carlo integral estimation.

基于传输的多保真预条件器用于马尔可夫链蒙特卡罗

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

本文提出一种基于传输的多保真预条件器，结合高保真与低保真模型，通过构造传输映射生成非高斯提议分布，实现高效采样，保证马尔可夫链的平稳分布为高保真后验。

Posterior Probabilities: Dominance and Optimism

The Bayesian posterior probability of the true state is stochastically dominated by that same posterior under the probability law of the true state. This generalizes to notions of "optimism" about posterior probabilities.

Gibbs posterior convergence and the thermodynamic formalism

In this paper we consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribu…

Mathlib概率库中的马尔可夫核

Markov kernels in Mathlib's probability library

本文探讨Mathlib概率库中马尔可夫核的定义与性质，介绍其在条件概率分布和后验分布中的应用，并讨论其在定义独立性、条件独立性和子高斯随机变量中的作用。

Bayesian Posterior Perturbation Analysis with Integral Probability Metrics

In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability of posterior measures in relation to perturbations in the likelihood potential and the prior measure. We present new stability results using a family of integral probability metrics (divergences) akin to dual problems that arise in optimal transport. Our results stand out from previous works in three directions: (1) We construct new families of integral prob…

On the Sequential Probability Ratio Test in Hidden Markov Models

We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.

Markov链尾概率的有限长度分析及其在简单假设检验中的应用

Finite-length Analysis on Tail probability for Markov Chain and Application to Simple Hypothesis Testing

本文利用信息几何推导有限长度Markov链样本均值尾概率及累积量生成函数的界，并据此建立简单假设检验中第二类错误最小概率的Hoeffding型界，还给出中心极限定理的新证明。