We consider generalized Bayesian inference on stochastic processes and dynamical systems with potentially long-range dependency. Given a sequence of observations, a class of parametrized model processes with a prior distribution, and a loss function, we specify the generalized posterior distribution. The problem of frequentist posterior consistency is concerned with whether as more and more samples are observed, the posterior distribution on parameters will asymptotically concentrate on the "right" parameters. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We show that the posterior distribution will asymptotically concentrate on parameters that minimize the expected loss and a divergence term, and we identify the divergence term as the Donsker-Varadhan relative entropy rate from process-level large deviations. As an application, we prove new quenched and annealed large deviation asymptotics and new Bayesian posterior consistency results for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, whenever estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

翻译：本文研究了具有潜在长程依赖性的随机过程与动力系统的广义贝叶斯推断问题。给定观测序列、带先验分布的参数化模型过程族以及损失函数，我们定义了广义后验分布。频率学派后验一致性关注的是：随着观测样本数量不断增加，参数的后验分布是否会在渐近意义下集中于"正确"参数上。我们证明，通过结合经典大偏差技术（如Varadhan引理、条件/淬火大偏差、退火大偏差及指数近似），可以推导出后验一致性。研究结果表明，后验分布将渐近集中于最小化期望损失与散度项的参数，其中散度项被识别为过程级大偏差的Donsker-Varadhan相对熵率。作为应用，我们为一类混合随机过程证明了新的淬火与退火大偏差渐近性及贝叶斯后验一致性结果。对于马尔可夫过程，当对数Sobolev常数估计可用时，可直接获得后验一致性的显式条件，这使得我们的框架本质上成为一个"黑箱"。此外，我们通过简洁的证明方法恢复了经典动力系统中的最新后验一致性结论。该框架为以统一方式证明各类贝叶斯过程的后验一致性提供了新的可能性。