Jon. Aaronson, Hitoshi Nakada
We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of "symmetric measure": exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.
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Let $\Om$ be a Borel subset of $S^\Bbb N$ where $S$ is countable. A measure is called exchangeable on $\Om$, if it is supported on $\Om$ and is invariant under every Borel automorphism of $\Om$ which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when $\Om=S^\Bbb N$. We apply the ergodic theory of equivalence relations to study the case $\Om\neq S^\Bbb N$, and obtain versions of this theorem when $\Om$ is a countable state Markov shift, and when $\Om$ is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint).
Hereditary subshifts whose measure of maximal entropy has no Gibbs property
本文证明遗传闭包的最大熵测度具有 Gibbs 性质当且仅当 Mirsky 测度纯原子，并给出 Keller 定理的逆定理。
By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type $α$ (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors.
Asymptotic behaviour of conditional $α$ diversity for the two-parameter Poisson-Dirichlet partition model and for the normalized generalized Gamma model has been recently investigated in Favaro et al. (2009, 2011) with a view to possible applications in Bayesian treatment of species richness estimation. Here we generalize those results to the larger class of mixed Poisson-Kingman species sampling models driven by the stable subordinator (Pitman, 2003).
We obtain some approximation results for the weights appearing in the exchangeable partition probability function identifying Gibbs partition models of parameter $α\in (0,1)$, as introduced in Gnedin and Pitman (2006). We rely on approximation results for central and non-central generalized Stirling numbers and on known results for conditional and unconditional $α$ diversity. We provide an application to an approximate Bayesian nonparametric estimation of discovery probability in species sampling problems under normalized inverse Gaussian priors.
The notion of Gibbs Measure is used by many researchers of the communities of Mathematical Physics, Probability, Thermodynamic Formalism, Symbolic Dynamics, and others. A natural question is when these several different notions of Gibbs measure coincide. We study the properties of Gibbs measures for functions with $d-$summable variation defined on a subshift $X$. Based on Meyerovitch's work, we prove that if $X$ is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expec…
Gibbs Measures on Subshifts of Finite Type: Five Equivalent Characterizations with Explicit Constants
本文基于拓扑混合有限型子移位，证明霍尔德势吉布斯测度的五种等价特征化，并以显式常数表达其等价性，涵盖转移算子谱间隙估计与统计极限定理。
Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpre…
Gibbs-type exchangeable random partitions, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially,…
Motivated by de Finetti's representation theorem for almost exchangeable arrays, we want to sample $\mathbf p \in [0,1]^d$ from a distribution with density proportional to $\exp(-A^2\sum_{i<j}c_{ij}(p_i-p_j)^2)$, where $A$ is large and $c_{ij}$'s are non-negative weights. We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every non-zero fixed matrix $C=(c_{ij})$, and large enough $A$, mixing happens in $Θ(A^2)$ steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix $C$ throu…
Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associa…
We study perturbations of topological pressures, Gibbs measures and measure-theoretic entropies of these measures concerning perturbed potentials defined on topologically transitive subshift of finite type. The subshift with respect to non-perturbed system is assumed to be no topologically transitive. Therefore, the subshift of the perturbed systems and the subshift of the unperturbed system are different. We reduce this situation to a perturbation problem of certain irreducible nonnegative matrices generated by Ruelle transfer operators. Consequently, under suitable conditions of potentials, …
Gibbs Measures on Multidimensional Spaces. Equivalences and a Groupoid Approach
本文证明了不同学科对子位移上吉布斯测度的多种定义在$d$-可和变差势下等价，并引入群胚方法建立了吉布斯测度与KMS态的等价关系。
For two collections of nonnegative and suitably normalised weights $\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,...,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $Π_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights $\W$ mu…
Asymptotic Inference for Exchangeable Gibbs Partitions
研究可交换吉布斯划分中参数估计和预测推断的渐近性质，通过混合正态分布推广了Ewens-Pitman模型的结果，并推导了预测概率简单形的渐近分布。
Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Recently those models have been investigated in a Bayesian nonparametric approach to species sampling problems as alternatives to the Dirichlet and the Pitman-Yor process priors. Here we derive marginals of conditional and unconditional multivariate distributions arising from exchangeable Gibbs partitions to obtain explicit formulas for joint falling factorial moments of corresponding c…
The distribution of age-ordered frequencies arising from an exchangeable Gibbs partition is studied in relation with the distribution of the positions at which new mutations appear in a sample.
Given a countable sofic group $Γ$, a finite alphabet $A$, a subshift $X \subseteq A^Γ$, and a potential $ϕ: X \to \mathbb{R}$, we give sufficient conditions on $X$ and $ϕ$ for expressing, in the uniqueness regime, the sofic entropy of the associated Gibbs measure $μ$ as the limit of the Shannon entropies of some suitable finite systems approximating $Γ\curvearrowright (X,μ)$. Next, we prove that if $μ$ satisfies strong spatial mixing, then the sofic pressure admits a formula in terms of the integral of a random information function with respect to any $Γ$-invariant Borel probability measure wi…
In this note, we point out that infinite-volume Gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a Hilbert space. This simple observation follows from a result of Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs measures can then be studied as single well-defined objects. This approach naturally extends the space of Random Overlap Structures as defined by Aizenman, Sims and Starr. We discuss the Ruelle Probability Cascades and the stochastic stability within this framework. As an application, we …
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