Gibbs posterior convergence and the thermodynamic formalism
Gibbs posterior convergence and the thermodynamic formalism
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DOI：
10.1214/21-aap1685
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发表时间：
2019-01
期刊：
影响因子：
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通讯作者：
K. Mcgoff;S. Mukherjee;A. Nobel
中科院分区：
文献类型：
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作者：
K. Mcgoff;S. Mukherjee;A. Nobel
在本文中，我们考虑了一个贝叶斯框架，用于从遍历观测中推断动力系统。所提出的贝叶斯过程基于吉布斯后验，这是标准贝叶斯推理的决策理论推广。我们在有限型混合位移上对由参数化吉布斯测度族组成的模型类设置先验。这个模型类通过允许长距离依赖（包括任意大阶的马尔可夫链）来推广（隐藏的）马尔可夫链模型。我们描述了Gibbs后验分布在参数空间上的渐近行为，即当观测数趋于无穷时。特别地，我们在模型系统与观测系统的连接空间上定义了一个极限变分问题，并证明了Gibbs后验分布集中在这个变分问题的解集周围。在适当指定模型的情况下，我们的收敛结果可以用来建立后验一致性。本研究建立了吉布斯后验推理与热力学形式的紧密联系，为相关过程的贝叶斯后验一致性研究提供了新的证明技术。
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 distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.