Consider a system consisting of a binary "stimulus" variable X, (e.g. whether an individual consumes tobacco), a binary "response" variable Y, (e.g. whether the individual develops a particular disease) and a "hidden" confounding variable U, (e.g. a hypothetical genetic factor that affects the individual's propensity to both consume tobacco as well as to develop the disease). Given P, the observed probability distribution of the pair (X, Y), we ask: what is the causal effect of X on Y? In particular, can it be determined solely from the knowledge of the "correlation" between X and Y? In the early 1990s, Judea Pearl proposed a formalization of the above question in the language of directed graphical models. Put differently, Pearl's framework formalized what it might mean to compute the strength of a causal relationship (such as may be measured in a controlled experiment) given only data about correlations among different components on the system.
It is intuitively clear that this problem is not always solvable: the example considered above is a case in point. However, a long line of work by several researchers culminated in 2006 in a complete algorithmic characterization of graphical models in which the problem is solvable. Further, this characterization also included an algorithmic procedure which takes the observed distribution and outputs the requisite "causal" distribution in those cases where the problem is solvable [Huang and Valtorta, 2006 and Shpitser and Pearl, 2006].
This talk will introduce directed graphical models and causal inference problem, and then give an overview of the solution of Huang-Valtorta and Shpitser-Pearl. We will then look at some recent progress on analyzing the 'robustness' (or 'condition number') of these solutions with respect to "noise" or "imperfections" in the description of the model or in the measurement of the observed distribution. Surprisingly, even though causal inference is a statistical problem, such robustness questions were only asked (and partly answered) recently, and several exciting future directions remain open. Time permitting, we will then look at other related notions of causal inference, such as the notion of directed mutual information in information theory, and the theory of linear structural equations.
Disclaimer: Part of this is joint work with Leonard J. Schulman (available from http://www.tifr.res.in/~piyush.srivastava/research.html#causal-inference). Some other parts are served with gentle dollops of caveats emptor.