Prob/Stat Seminar

Simulations of stochastic neuron potential models, which describe the voltage potential along the length of a neuron’s axon and incorporate ion channel noise as Gaussian fluctuations, have shown that channel noise can induce complex phenomena such as jitters and splitting of action potentials [1] and place constraints on the miniaturization of axons [2]. To develop a robust analytic framework for understanding stochastic effects of channel noise on action potential propagation in a neuron, we need to begin by investigating how many, independent and spatially distributed ion channels can collectively yield deterministic behavior. We start with an electrophysiological derivation of a simple discrete model and contrast this with a common, yet less physically accurate approach where the law of large numbers and the central limit theorem are more easily applied. Our model couples a spatially discretized diffusive PDE for the voltage with continuous-time Markov processes that govern the behavior of the ion channels. We will then outline an argument using homogenization theory to estimate the rate of strong convergence to the typical deterministic PDE as the spacing between ion channels approaches zero. Finally, we present a numerical technique for simulating our model and discuss the challenges involved in increasing computational efficiency of simulations. [1] Faisal AA, Laughlin SB. Stochastic simulations on the reliability of action potential propagation in thin axons. PLoS Comput Biol. 2007 May;3(5):e79. doi: 10.1371/journal.pcbi.0030079. PMID: 17480115; PMCID: PMC1864994. [2] Faisal AA, White JA, Laughlin SB. Ion-channel noise places limits on the miniaturization of the brain's wiring. Curr Biol. 2005 Jun 21;15(12):1143-9. doi: 10.1016/j.cub.2005.05.056. PMID: 15964281.

Langevin dynamics is Newton's law for the motion of N particles subject to friction, thermal fluctuations and potential forces. Aside from its relevance in statistical mechanics, its discretizations are used in Markov chain Monte Carlo to draw samples from its explicit, and moldable, stationary distribution by running the system long enough. Because of its ballistic, as opposed to diffusive, behavior, it is believed to have a better rate of convergence to equilibrium when compared to stochastic gradient dynamics (also known as "overdamped Langevin"). However, the precise mechanisms leading to geometric ergodicity of the system are more nuanced than stochastic gradient dynamics, especially because the SDE is degenerate elliptic and damping only explicitly acts on the momentum directions. This has led to an abundance of research on the topic. The goal of this talk is to give an overview of methods used to establish convergence to equilibrium for Langevin dynamics forced by a wide class of potential functions. In the process, we will give an overview of results. Particular attention will be paid to both probabilistic and functional analytic methods.

Extreme value theory for stationary processes has been well developed over the past decades. Central to the theory is the notion of the extremal index, which quantifies the strength of clustering of extreme values due to dependence and appears as a correcting factor in the limit distribution. Classical theory provides a predictive formula for the extremal index, known as the candidate extremal index, which is typically equal to the extremal index under mild regularity conditions. In this talk, we introduce a double series model constructed via intersections of independent discrete renewal processes with infinite mean. This model exhibits an unusual phenomenon: the extremal index and the candidate extremal index differ. We further discuss several other surprising consequences that arise from this discrepancy. The talk is based on joint works with Rafal Kulik and Yizao Wang.

I will start with a self-contained introduction to the homogenization of inviscid (first-order) and viscous (second-order) Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension and then give a survey of the now-classical works that are concerned with periodic media or convex Hamiltonians. Afterwards, I will drop both of these assumptions and outline the results obtained in the last decade that: (i) established homogenization for inviscid HJ equations in one dimension; and (ii) provided counterexamples to homogenization in the inviscid and viscous cases in dimensions two and higher. Finally, I will present my recent joint work with E. Kosygina in which we prove homogenization for viscous HJ equations in one dimension, and also describe how the solution of this problem differs qualitatively from that of its inviscid counterpart.

Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. While GPs are commonly used for prediction, in certain applications, the focus is on parameter inference. To illustrate the importance of parameter inference, we examine a specific application of GPs in enhancer-promoter time series. We then discuss the identifiability theory of Gaussian process parameters in the Euclidean domain and extensions to compact Riemannian manifold domains.

Many scientific problems involve data exhibiting both temporal and cross-sectional dependencies. While linear models for such data are well understood, the theoretical analysis of nonlinear estimators remains comparatively scarce. In this talk, I will discuss a kernel-based estimation procedure for nonlinear dynamics within the reproducing kernel Hilbert space framework, focusing on nonlinear stochastic regression and nonlinear vector autoregressive models. The main results establish nonasymptotic probabilistic bounds on the deviation between the kernel estimator and the true nonlinear regression function.

Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.

This talk presents a framework for obtaining statistical guarantees for subgraph densities of a general population network under without-replacement sampling (SRSWOR). We establish the asymptotic normality of the Horwitz-Thompson (HT) estimator and propose a jackknife variance estimator. We also analyze joint asymptotic normality of multiple subgraph densities and extend to multi-edge networks, with applications to graph hypothesis testing.

We propose a novel Bayesian methodology—Bayesian causal synthesis—to mitigate model misspecification and improve estimation of heterogeneous treatment effects. This approach identifies a synthesis function that minimizes bias from unobserved confounding. We demonstrate the method’s consistency and performance through simulations and an empirical study.

We propose a new model for generating and estimating unweighted networks that allows for flexible community structures and likelihood-based estimation. This model provides a broader class of network representations, improving estimation of nodal features and insights into heterogeneous network structures.

This talk reviews recent work on Weighted Ensemble, a method for enhanced Monte Carlo sampling. The method involves interacting Markov chains to efficiently sample rare events. Results are based on collaborations with D. Aristoff, R.J. Webber, and D.M. Zuckerman.