10th World Congress in Probability and Statistics

Invited Session (live Q&A at Track 1, 11:30AM KST)

Invited 04

Mathematical Population Genetics and Computational Statistics (Organizer: Paul Jenkins)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 19 Mon, 10:30 PM — 11:00 PM EDT

Mapping genetic ancestors

Graham Coop (University of California at Davis)

2
Spatial patterns in genetic diversity are shaped by the movements of individuals dispersing from their parents and populations expanding and contracting. It has long been appreciated that these patterns of movement leave shape the underlying genealogies along the genome leading to geographic patterns of isolation by distance in contemporary population genetic data. The enormous amount of information contained in genealogies along recombining sequences has, up till now, not been amenable to this approach. However, it is now possible to infer a sequence of gene genealogies along a recombining sequence. Here we capitalize on this important advance and develop methods to use thousands of trees to estimate time-varying per-generation dispersal rates and to locate the genetic ancestors of a sample back through time. We take a likelihood approach using a simple approximate spatial model (Branching Brownian Motion) as our prior distribution of genealogies. After testing our method with simulations we apply it to the 1001 Genomes dataset of over one thousand Arabidopsis thalianagenomes sampled across a wide geographic extent. We detect a very high dispersal rate in the recent past, especially longitudinally, and use inferred ancestor locations to visualize many examples of recent long-distance dispersal and recent admixture events. We also use inferred ancestor locations to identify the origin and ancestry of the North American expansion, to depict alternative geographic ancestries stemming from multiple glacial refugia. Our method highlights the huge amount of largely untapped information about past dispersal events and population movements contained in genome-wide genealogies.

Cellular point processes: quantifying cell signaling

Barbara Engelhardt (Princeton University)

3
This talk does not have an abstract.

Fitting stochastic epidemic models to gene genealogies using linear noise approximation

Vladimir Minin (University of California, Irvine)

4
Phylodynamics is a set of population genetics tools that aim at reconstructing demographic history of a population based on molecular sequences of individuals sampled from the population of interest. One important task in phylodynamics is to estimate changes in (effective) population size. When applied to infectious disease sequences such estimation of population size trajectories can provide information about changes in the number of infections. To model changes in the number of infected individuals, current phylodynamic methods use non-parametric approaches (e.g., Bayesian curve-fitting based on change-point models or Gaussian process priors), parametric approaches (e.g., based on differential equations), and stochastic modeling in conjunction with likelihood-free Bayesian methods. The first class of methods yields results that are hard to interpret epidemiologically.
The second class of methods provides estimates of important epidemiological parameters, such as infection and removal/recovery rates, but ignores variation in the dynamics of infectious disease spread. The third class of methods is the most advantageous statistically, but relies on computationally intensive particle filtering techniques that limits its applications. We propose a Bayesian model that combines phylodynamic inference and stochastic epidemic models, and achieves computational tractability by using a linear noise approximation (LNA) --- a technique that allows us to approximate probability densities of stochastic epidemic model trajectories. LNA opens the door for using modern Markov chain Monte Carlo tools to approximate the joint posterior distribution of the disease transmission parameters and of high dimensional vectors describing unobserved changes in the stochastic epidemic model compartment sizes (e.g., numbers of infectious and susceptible individuals). We illustrate our new method by applying it to Ebola genealogies estimated using viral genetic data from the 2014 epidemic in Sierra Leone and Liberia.

Q&A for Invited Session 04

0
This talk does not have an abstract.

Session Chair

Paul Jenkins (University of Warwick)

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Invited 18

Deep Learning (Organizer: Johannes Schmidt-Hieber)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 19 Mon, 10:30 PM — 11:00 PM EDT

Dynamics and phase transitions in deep neural networks

Yasaman Bahri (Google Research)

5
The study of deep neural networks whose hidden layer widths are large has been fruitful in building theoretical foundations for deep learning. I will begin by surveying the result of our past work along these lines. For instance, infinitely-wide deep neural networks can be exactly described by Gaussian processes with particular compositional kernels, both in their prior and predictive posterior distributions. Furthermore, such infinite-width deep networks can be exactly described as linear models under gradient descent up to a maximum learning rate. At larger learning rates with squared loss, empirical evidence suggests a phase transition to a different, nonlinear regime with universal features across architectures and datasets. I will describe our theoretical understanding of this phase transition through the study of a class of simple dynamical systems distilled from neural network evolution in function space.

Theoretical understanding of adding noises to deep generative models

Yongdai Kim (Seoul National University)

9
Deep generative models have received much attention recently since they can generate realistic synthetic images. Recently, some researches have reported that adding noises to data is helpful to learn deep generative models. In this talk, we provide theoretical justifications of this method. We derive the convergence rate of the maximum likelihood estimator of a deep generative model and show that the convergence rate can be improved by adding noises in particular when the noise level of data is small.

Adversarial examples in random deep networks

Peter Bartlett (University of California at Berkeley)

5
Because the phenomenon of adversarial examples in deep networks poses a serious barrier to the reliable and robust application of this methodology, there has been considerable interest in why it arises. We consider ReLU networks of constant depth with independent gaussian parameters, and show that small perturbations of input vectors lead to large changes of outputs. Building on insights of Daniely and Schacham (2020) and of Bubeck et al (2021), we show that adversarial examples arise in these networks because the functions that they compute are very close to linear. The main result is for networks with constant depth, but we also show that some constraint on depth is necessary for a result of this kind: there are suitably deep networks that, with constant probability, compute a function that is close to constant.

Joint work with S├ębastien Bubeck and Yeshwanth Cherapanamjeri

Q&A for Invited Session 18

0
This talk does not have an abstract.

Session Chair

Johannes Schmidt-Hieber (University of Twente)

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