Attributing extreme-event probability to a source variable

Abstract

Information-flow theory quantifies directional coupling through the rate of change of a target's Shannon entropy, a bulk functional insensitive to the tail of the distribution. We ask instead how a source variable contributes to the probability that the target exceeds a threshold. For source-additive drift the source's share of the marginal probability current is exact, and it separates into a mean-forcing part and a conditional-excess part that vanishes under independence. An identity links the two descriptions: the Liang information flow is the density-weighted mean of the derivative of the specific source current, whereas the exceedance current is its level at the threshold. This explains why entropy-based coupling collapses in saturated regimes where the source contributes most to the extreme; the Rényi information flow, a higher-order expansion and a Fisher-normalised response fail for the same reason. The flux decomposition, although exact, does not attribute: its terms are gross transports that nearly cancel. The quantity that does attribute is an adjoint response built from the backward generator which, read as a relative change, is uniformly accurate across two decades of event probability. We map the operating envelope of the estimator under omitted drivers, hidden slow memory, state-dependent coupling and multiplicative noise. Under correct specification the attribution carries a reproducible shortfall of ten to twenty-five per cent; the misspecifications we test bias it upward by up to ninety per cent. Applied to the 2003 European and 2010 Russian heatwaves in reanalysis, the attributed contribution of soil moisture is strongly threshold dependent, rising from the per cent level at moderate thresholds to a factor of two at the rarest, so a contribution quoted without its threshold is underspecified.