Traditional statistical measures of causal effect, such as the average causal effect, compare outcome values under different cause values. Information-theoretic measures, such as mutual information, compare outcome distributions and thus can capture nonlinear interactions and effects on variance. However, they are defined for a random variable as a whole and do not distinguish the contribution of specific cause values. This work fills that gap by introducing measures that retain the flexibility of the information-theoretic approach but are tied to a specific cause value, allowing questions like "What is the effect of El Niño specifically, rather than ENSO as a whole?" In the context of neighboring work in machine learning and numerical climate modeling, this study stands apart: it translates the apparatus of causal inference into the language of information theory.
The authors formalize measures of specific total, direct, and indirect effects via the Kullback-Leibler divergence between the outcome distribution after an intervention and the "natural" distribution averaged over all possible cause values. For identifiability from observational data, they use Pearl's do-calculus. As an application, they analyze nine runs of the CESM2 climate model over 1850–2015. ENSO is quantized into three phases (El Niño, La Niña, neutral) based on the Niño 3.4 index, and Pacific Northwest temperature is averaged over two-week intervals and quantized into three categories (cold, warm, neutral anomaly). All distributions are estimated by maximum likelihood, and significance is tested with a permutation bootstrap test.
Estimates of the normalized measures show that the specific total effect of the El Niño phase on temperature is about three times the average effect across all phases. The direct effect is larger than the indirect effect for all cause values, consistent with the physical mechanism: ENSO influences temperature primarily directly through atmospheric circulation, not through temperature persistence. The most pronounced effect is found for the conditional influence of past temperature on current temperature during the El Niño phase: an intervention setting a cold anomaly in an El Niño year yields an effect roughly five times stronger than average. All estimates are statistically significant, but confidence intervals for the indirect effect include zero for all phases except El Niño.
The work shows that information-theoretic measures can be adapted to answer questions about specific cause values while retaining their ability to capture nonlinear effects. This brings together two approaches to causal inference and opens the way for more detailed analysis in climatology and neuroscience. Unlike neighboring work that either uses machine learning to predict properties or applies classical statistical methods, here a rigorous theoretical framework is proposed and tested on real climate data.
The method can be extended to continuous variables and to the case of partially observed covariates. It may also be possible to learn causal graphs from data and then apply specific measures. In climatology, promising directions include studying other regions and variables, such as precipitation or pressure, as well as analyzing longer time series.
The proposed measures could be useful in neuroscience for assessing the influence of specific stimuli on neural activity, in epidemiology for analyzing the effects of rare exposures, and in economics for evaluating the impact of specific policy decisions.
The next step could be developing estimators for continuous distributions and methods for choosing quantization thresholds. It is also important to formally analyze the statistical properties of the proposed estimators and tests.
In climatology, the question of separating direct atmospheric and indirect (via ocean, soil) mechanisms of ENSO's influence on regional climate remains open. The proposed method allows quantifying the contribution of each pathway, which could help improve climate models.
The mutual information between winning the lottery and spending is nearly zero, because the probability of winning is minuscule, although for a specific winner the effect is huge. Specific measures capture this contrast.