Identifiability of causal graphs under nonadditive conditionally parametric causal models

Causal discovery from observational data is a very challenging, often impossible, task. However, estimating the causal structure is possible under certain assumptions on the data-generating process. Many commonly used methods rely on the additivity of the noise in the structural equation models. Additivity implies that the variance or the tail of the effect, given the causes, is invariant; the cause only affects the mean. In many applications, it is desirable to model the tail or other characteristics of the random variable since they can provide different information about the causal structure. However, models for causal inference in such cases have received only very little attention. It has been shown that the causal graph is identifiable under different models, such as linear non-Gaussian, post-nonlinear, or quadratic variance functional models. We introduce a new class of models called the Conditional Parametric Causal Models (CPCM), where the cause affects the effect in some of the characteristics of interest.We use the concept of sufficient statistics to show the identifiability of the CPCM models, focusing mostly on the exponential family of conditional distributions.We also propose an algorithm for estimating the causal structure from a random sample under CPCM. Its empirical properties are studied for various data sets, including an application on the expenditure behavior of residents of the Philippines.