Probability and Statistics Seminar

Sequential Monte Carlo (SMC) algorithms find applications across a wide range of areas, most notably in non-linear filtering and smoothing for state space models. The performance of SMC is limited, particularly in smoothing applications, by the phenomenon of ancestral degeneracy, an inevitable consequence of the resampling step. We study the genealogies induced by resampling, showing that the genealogies of many popular SMC algorithms converge to Kingman’s n-coalescent. The tractability of this limiting process allows ancestral degeneracy to be quantified and thus hopefully mitigated, by informing the choice of algorithm and tuning parameters. Joint work with Paul Jenkins, Adam Johansen and Jere Koskela.

We consider the (1+1)-dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar-Parisi-Zhang (KPZ) equation. We show an exact way of computing all the positive real Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial data. This is a joint work with Promit Ghosal.

The directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble, give an extension of this construction for optimal length disjoint paths in the directed landscape, and show how these constructions reveal surprising Brownian structures in the directed landscape. Based on joint work with J. Ortmann, B. Virag, and L. Zhang.

In this talk, we will present a simplified second-order Gaussian Poincaré inequality for the normal approximation of functionals over infinitely many Rademacher random variables. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method. As an application, the number of vertices with prescribed degree in the Erdös-Rényi random graph is discussed. The same statistic is also studied for the percolation on the Hamming hyperbube.

In statistical physics many interesting phenomena, e.g. behavior of systems at critical parameters or in the theory of hydrodynamic limits, arise from systems having multiple time-scales. A slow component is influenced by fast components, and as the number of interacting components tends to infinity, limiting results for the slow component are obtained in terms of `averaged' versions of the fast components.

I will consider in my talk the large deviations of coupled Markovian systems with two-time scales. These fluctuations can arise from two sources: fluctuations of the slow process itself, or fluctuations of the large time averages of the fast process, effectively leading to a competition of two fluctuation effects. To obtain the large deviation principle, we consider an associated Hamilton-Jacobi-Bellman equation of which the Hamiltonian is given in terms of the two fluctuation effects. We establish under mild conditions that this Hamilton-Jacobi-Bellman equation is well-posed, and as a consequence that we have a large deviation principle for a wide class of weakly coupled Markov processes.

Based on joint work with Mikola Schlottke (Eindhoven, The Netherlands).

We consider statistical inference for a class of agent-based SIS and SIR models. In these models, agents infect one another according to random contacts made over a social network, with an infection rate that depend on individual attributes. Infected agents might recover according to another random mechanism that also depends on individual attributes, and observations might involve occasional noisy measurements of the number of infected agents. Likelihood-based inference for such models presents various computational challenges. In this talk, I will present various Monte Carlo algorithms to address these challenges.

We introduce the stochastic Ricci flow in two spatial dimensions, which is a stochastic partial differential equation. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on compact surfaces, in the so called "L1 regime".

Inferring correlation among multiple continuous and discrete biological traits along an evolutionary history remains an important yet challenging problem. We jointly model these mixed-type traits through data augmentation and a phylogenetic multivariate probit model. With large sample sizes, posterior computation under this model is problematic, as it requires repeated sampling from a high-dimensional truncated Gaussian distribution with strong correlation. For this task, we propose the Hamiltonian zigzag sampler based on Laplace momentum, one state-of-the-art Markov chain Monte Carlo method.

The reversible Hamiltonian zigzag sampler achieves better efficiency than its non-reversible competitors including the Markovian zigzag sampler and the bouncy particle sampler that is the best current approach for sampling latent parameters in the phylogenetic probit model. In an application with 535 HIV viruses and 24 traits that necessitates sampling from a 12,840-dimensional truncated normal, our method makes it possible to estimate the across-trait correlation and detect association between immune escape mutations and the pathogen’s capacity to cause disease.

Many stochastic particle systems have well-defined continuum limits: as the number of particles tends to infinity, the density of particles converges to a deterministic limit that satisfies a partial differential equation. In this talk I will discuss a specific example of this: a system consisting of particles with finite size. In two and three dimensions they are spheres, in one dimension rods. The particles move by Brownian noise and cannot overlap with each other, leading to a strong interaction with neighbouring particles. Previous studies include numerical simulations and formal asymptotic results, along with conjectures on the limit, but no rigorous results.

We will consider the one-dimensional setting and a scaling in which the number of particles tend to infinity while the volume fraction of the rods remain constant. Using large deviations for empirical measures we give a complete picture of the convergence of the particle system and derive the gradient flow structure for the limit evolution. The latter gives clear interpretations for the driving functional and the dissipation metric and how they derive from the underlying particle system.

This is based on joint work with Nir Gavish and Mark Peletier.