Graphical models can represent multivariate distributions in an intuitive way and, hence, facilitate statistical analysis of high-dimensional data. Such models are usually modular so that high-dimensional distributions can be described and handled by careful combination of lower dimensional factors. Furthermore, graphs are natural data structures for algorithmic treatment. Moreover, graphical models can allow for causal interpretation, often provided through a recursive system on a directed acyclic graph (DAG) and the max-linear Bayesian network we introduced in [1] is a specific example. This talk contributes to the recently emerged topic of graphical models for extremes, in particular to max-linear Bayesian networks, which are max-linear graphical models on DAGs.

In this context, the Latent River Problem has emerged as a flagship problem for causal discovery in extreme value statistics. In [2] we provide a simple and efficient algorithm QTree to solve the Latent River Problem. QTree returns a directed graph and achieves almost perfect recovery on the Upper Danube, the existing benchmark dataset, as well as on new data from the Lower Colorado River in Texas. It uses pariwise dependence, handles missing data, and has an automated parameter tuning procedure. In our paper, we also show that, under a max-linear Bayesian network model for extreme values with propagating noise, the QTree algorithm returns asymptotically a.s. the correct tree. Here we use the fact that the non-noisy model has a left-sided atom for every bivariate marginal distribution, when there is a directed edge between the the nodes.

[1] Gissibl, N. and Klüppelberg, C. (2018). Max-linear models on directed acyclic graphs. Bernoulli 24(4A), 2693-2720.

[2] Ngoc, M.T., Buck, J., and Klüppelberg, C. (2021). Estimating a latent tree for extremes. Submitted, arXiv:2102.06197.