Measuring Latent Causal Structure

Discovering latent representations of the observed world has become increasingly more relevant in the artificial intelligence literature [Hinton and Salakhutdinov, 2006, Bengio and Cun, 2007]. Much of the effort concentrates on building latent variables which can be used in prediction problems, such as classification and regression. A related goal of learning latent structure from data is that of identifying which hidden common causes generate the observations. This becomes relevant in applications that require predicting the effect of policies. As an example, consider the problem of identifying the effects of the “industrialization level” of a country on its “democratization level” across two different time points. Democratization levels and industrialization levels are not directly observable: they are hidden common causes of observable indicators which can be recorded and analyzed. For instance, gross national product (GNP) is an indicator of industrialization level, while expert assessments of freedom of press can be used as indicators of democratization. Extended discussions on the distinction between indicators and the latent variables they measure can be found in the literature of structural equation models [Bollen, 1989] and errorin-variables regression [Carroll et al., 1995]. Causal networks can be used as a language to represent this information. We postulate a graphical encoding of causal relationships among random variables, where vertices in the graph representing random variables and directed edges Vi → Vj represent the notion that Vi is a direct cause of Vj . Formal definitions of direct causation and causal networks are given by Spirtes et al. [2000] and Pearl [2000]. In our setup, we explicitly represent latent variables of interest as vertices in the graph. For example, in Figure 1 we have a network representation for the problem of causation between industrialization and democratization levels. This model makes assumptions about the connections among latent variables themselves: e.g., industrialization causes democratization, and the possibility of