We study the Bayesian linear model $ L = A^T D + \epsilon $ where $\epsilon \sim iid N(0,\sigma^2)$ and $\epsilon$ is independent of $D$ . The vector $D$ is very high-dimensional, random and `outlier prone' so that it frequently contains a very few untypically large elements. We are particularly interested in the case where $A$ is seriously rank deficient, so that ordinarily there is no hope of estimating $D$ itself. We aim, not to recover it, but in the event that $L$ contains extreme deviations from typical behavior, to estimate only the components of $D$ that are atypically large. In a sense, we are seeking to identify the hypothesized sparse causes of extreme events, when extreme events occur.

A possible application area for our tools is in network `traffic matrix problem'. We suggest that this approach might prove useful in detecting `flash crowds' and other anomalies from link-level traffic measurements.

In this paper, we formalize the problem of inferring sparse causes for extreme events in a decision-theoretic fashion, develop a bayesian model for it, and an inference procedure based essentially on monte-carlo variable selection using the Gibbs sampler as in Foster and George. We discuss the application in a network setting, and work out several simple examples.