Week 28.01.2024 – 03.02.2024

The contact process is a model for the spread of an infection in a graph. Vertices can be either healthy or infected\DSEMIC infected vertices recover with rate 1 and send the infection to each neighbor with rate lambda. A key question of interest is: if we start the process with a single infected vertex, can the infection survive forever with positive probability? This typically depends on the graph and on the value of lambda\DSEMIC for instance, on integer lattices, there is a critical value of lambda at which the survival probability changes from zero to strictly positive. However, on graphs that include vertices of high degree, such as Galton-Watson trees with heavy-tailed offspring distributions, it has been observed that the infection survives with positive probability for all values of lambda, no matter how small. This is because high-degree vertices sustain the infection for a long time and send the infection to each other. In this work, we investigate this survival-for-all-lambda phenomenon for a modification of the contact process, which we introduce and call the penalized contact process. In this new process, vertex u transmits the infection to neighboring vertex v with rate lambda/max(degree(u),degree(v))^mu, where mu>0 is an additional parameter (called the penalization exponent). This is inspired by considerations from social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that the introduction of this penalty factor introduces a rich range of behavior for the phase diagram of the contact process on Galton-Watson trees. We also show corresponding results for the penalized contact process on finite graphs obtained from the configuration model.

Joint work with Júlia Komjáthy and Zsolt Bartha.

We describe the constructions of implosion and contraction for
complex-symplectic or hyperkahler manifolds. Implosions are examples
of nonreductive quotients in geometric invariant theory. We can
interpret both constructions in terms of a generalisation of the
Moore-Tachikawa category that was introduced in physics.

In this talk I will discuss the Nevanlinna-Pick interpolation problem under indeterminacy conditions. In terms of rational functions, the first and second kind, we obtain the explicit formula for the Nevanlinna matrix. The solutions to the interpolation problem are described through linear fractional transformations including Nevanlinna functions. This is a direct analog of the Nevanlinna formula for the Hamburger moment problem.

Many statistical problems in causal inference involve a probability distribution other than the one from which data are actually observed\DSEMIC as an additional complication, the object of interest is often a marginal quantity of this other probability distribution. This creates many practical complications for statistical inference, even where the problem is non-parametrically identified. In particular, it is difficult to perform likelihood-based inference, or even to simulate from the model in a general way.

We introduce the frugal parameterization, which places the causal effect of interest at its centre, and then builds the rest of the model around it. We do this in a way that provides a recipe for constructing a regular, non-redundant parameterization using causal quantities of interest. In the case of discrete variables we can use odds ratios to complete the parameterization, while in the continuous case copulas are the natural choice. Our methods allow us to construct and simulate from models with parametrically specified causal distributions, and fit them using likelihood-based methods, including fully Bayesian approaches. Our proposal includes parameterizations for the average causal effect and effect of treatment on the treated, as well as other common quantities of interest.

I will also discuss some other applications of the frugal parameterization, including to survival analysis, generative modelling, parameterizing nested Markov models, and ‘Many Data’: combining randomized and observational datasets in a single parametric model.

This is joint work with Vanessa Didelez (University of Bremen and BIPS), Xi Lin and Daniel Manela (both Oxford).

Reference
Evans, R.J. and Didelez, V. Parameterizing and Simulating from Causal Models (with discussion), J. Roy. Statist. Ser. B, 2024.