Browsing by Author "Su, Langxuan"

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    A Large Deviation Approach to Posterior Consistency in Dynamical Systems
    Su, Langxuan; Mukherjee, Sayan
    In this paper, we provide asymptotic results concerning (generalized) Bayesian inference for certain dynamical systems based on a large deviation approach. Given a sequence of observations $y$, a class of model processes parameterized by $\theta \in \Theta$ which can be characterized as a stochastic process $X^\theta$ or a measure $\mu_\theta$, and a loss function $L$ which measures the error between $y$ and a realization of $X^\theta$, we specify the generalized posterior distribution $\pi_t(\theta \mid y)$. The goal of this paper is to study the asymptotic behavior of $\pi_t(\theta \mid y)$ as $t \to \infty.$ In particular, we state conditions on the model family $\{\mu_\theta\}_{\theta \in \Theta}$ and the loss function $L$ such that the posterior distribution converges. The two conditions we require are: (1) a conditional large deviation behavior for a single $X^\theta$, and (2) an exponential continuity condition over the model family for the map from the parameter $\theta$ to the loss incurred between $X^\theta$ and the observation sequence $y$. The proposed framework is quite general, we apply it to two very different classes of dynamical systems: continuous time hypermixing processes and Gibbs processes on shifts of finite type. We also show that the generalized posterior distribution concentrates asymptotically on those parameters that minimize the expected loss and a divergence term, hence proving posterior consistency.
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    Absolute Continuity of Singular SPDEs and Bayesian Inference on Dynamical Systems
    (2023) Su, Langxuan

    We explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

    In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

    In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. 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 identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when 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.

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    The Gaussian Structure of the Singular Stochastic Burgers Equation
    Mattingly, Jonathan C; Romito, Marco; Su, Langxuan
    We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed as we are beyond the regime of classical solutions for much of the paper.