Graphical models are a popular tool to analyse and visualise dependence properties between random variables; see e.g. Lauritzen (1996). Each node in the graph represents a random variable, and the absence of an edge between two nodes indicates conditional independence between the corresponding random variables. We introduce a new recursive structural equation model, where all random variables can be written as a max-linear function of their parents and independent noise variables. 

In particular, we define directed graphical models, where edge orientations come along with an intuitive causal interpretation and present conditional independence properties of the model. Furthermore, we use algebraic methods to characterise all max-linear models, which are generated by a structural equation model and detail the relation between the coefficients of the structural equation model (the edge weights of the graph) and the max-linear coefficients. Finally, we extend finite-dimensional max-linear models to models on infinite graphs, and investigate their relations to classical percolation theory, more precisely to nearest neighbour Bernoulli bond percolation. The talk is based on joint work with Nadine Gissibl, Steffen Lauritzen, Moritz Otto, and Ercan Sönmetz.

About the Speaker

Claudia Klüppelberg is a Full Professor for Mathematical Statistics at the Technical University of Munich. After studying mathematics and receiving her doctorate (1987) at the University of Mannheim, she completed her Habilitation at ETH Zurich (1993). Her research interests lies in probabilistic risk modelling and the development of new methods for risk assessment. At present her work focuses on risk spreading in networks. Along with more than 150 publications in scientific journals, she has edited various books, and written the book "Modelling Extremal Events for Insurance and Finance (jointly with Paul Embrechts and Thomas Mikosch). She is an Elected Fellow of the Institute of Mathematical Statistics and President-Elect of the Bernoulli Society.