伊顿马尔可夫链何时不可约？

用推广的Eaton马尔可夫链评估违约先验

Evaluating Default Priors with a Generalization of Eaton's Markov Chain

我们考虑评估不适当的先验在一个正式的贝叶斯设置根据其使用的后果。设$Φ$是参数空间上的一类函数，考虑二次损失下$Φ$的元素估计.如果$Φ$中每个函数的形式Bayes估计是可容许的，则先验关于$Φ$是强可容许的.伊顿建立强容许性的方法是基于研究与推理设置相关的特定马尔可夫链的稳定性。在以前的工作中，根据$ϕ\in Φ$是有界的还是无界的，对此进行了不同的处理。我们介绍和研究一个新的马尔可夫链，使我们能够统一和推广现有的方法，同时扩大其潜在的适用范围。为了说明该方法，我们建立强容许条件时，模型是一个$p$维多元正态分布与未知的均值向量$θ$和先验的形式是$ν(\|θ\|^{2})dθ$。

On the use of Markovian stick-breaking priors

In [10], a `Markovian stick-breaking' process which generalizes the Dirichlet process $(μ, θ)$ with respect to a discrete base space ${\mathfrak X}$ was introduced. In particular, a sample from from the `Markovian stick-breaking' processs may be represented in stick-breaking form $\sum_{i\geq 1} P_i δ_{T_i}$ where $\{T_i\}$ is a stationary, irreducible Markov chain on ${\mathfrak X}$ with stationary distribution $μ$, instead of i.i.d. $\{T_i\}$ each distributed as $μ$ as in the Dirichlet case, and $\{P_i\}$ is a GEM$(θ)$ residual allocation sequence. Although the motivation in [10] was to rela…

马尔可夫基因谱过程的序列学习理论

Sequential learning theory for Markov genealogy processes

本文基于滤子框架，探讨了添加物种如何提升系统动力学推断，并通过分解方差减少成分，揭示了学习类估计量与吸收估计量之间的本质差异，指出序列数据难以完全揭示潜在基因谱信息。

不可约量子马尔可夫链的渐近统计理论

Asymptotic statistical theory of irreducible quantum Markov chains

本文研究不可约量子马尔可夫链的渐近统计理论，探讨其可识别性及渐近收敛性。通过分析可识别参数空间的几何性质，揭示周期性与结构的关系，并基于orbifold图表建立局部渐近理论。

Learning from non-irreducible Markov chains

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this artic…

Information Geometry of Reversible Markov Chains

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions…