A Logic for Causal Inference in Time Series with Discrete and Continuous Variables Samantha Kleinberg Columbia University New York, NY
[email protected] Abstract windows of time. Similarly, Granger causality [1969] allows for only discrete lags between cause and effect and assumes Many applications of causal inference, such as find- that the relationships are between individual variables. Fi- ing the relationship between stock prices and news nally, prior work on inferring complex causal relationships reports, involve both discrete and continuous vari- represented as logical formulas identified factors that substan- ables observed over time. Inference with these tially impact the probability of the effect, but required that all complex sets of temporal data, though, has re- variables be discretized [Kleinberg and Mishra, 2009]. mained difficult and required a number of simplifi- In this work we address the problem of inference when we cations. We show that recent approaches for infer- are interested primarily in the level of the effect. We will ring temporal relationships (represented as logical show that instead of a difference in probability, a cause’s sig- formulas) can be adapted for inference with contin- nificance for an effect can be assessed using an average dif- uous valued effects. Building on advances in logic, ference in conditional expectation. By extending the under- PCTLc (an extension of PCTL with numerical con- lying logic used to represent the relationships, this approach straints) is introduced here to allow representation allows for structured representation and automated inference and inference of relationships with a mixture of of complex temporal relationships with discrete and contin- discrete and continuous components.