Identifying causal directions between mixed categorical and continuous data nodes
On the Identifiability of Mixed Ordinal and Exponential Family Causal DAGs under Linear Parametric Models
Machine Learning
Summary
This paper studies how to tell which way cause-and-effect relationships go between variables that are mixed types—specifically, some variables are ordinal (with a set order) and others follow a special mathematical family called the exponential family. The authors show that if the ordinal variable has at least three levels and the other variable has at least three distinct values, the direction of influence between these two variables can be identified just from the data. They also prove these conditions are necessary in certain cases, and demonstrate how their results help find causal directions that standard methods cannot.
What this means in practice
- •For data scientists: Determine causal directions in complex datasets mixing ordered categories and exponential family variables beyond what independence tests can reveal.
- •For machine learning engineers: Improve causal structure learning algorithms by incorporating mixed-type models that guarantee edge orientation identifiability under general conditions.
Authors
Sambit Mishra, Urbashi Mitra
Abstract
The problem of identifiability in linear parametric models (LPMs) whose nodes follow either an ordered logit model or a regular one-parameter exponential family is evaluated. The results go beyond classical structural equation models as well as results for nodes with observations from a homogeneous family of distributions. The main result establishes that the orientation of every edge joining an ordinal node to an exponential-family node is identifiable from the joint distribution alone at every parameter value, provided the ordinal node has at least three categories and the exponential-family node at least three points of support, with no restriction on the sufficient statistic. Converses show that both requirements are necessary: the three-category requirement is binding only for affine sufficient statistics, and the three-point requirement is binding under the canonical link. The guarantee extends to orienting every such mixed ordinal-exponential family edge of a given $d$-node undirected skeleton. Numerical experiments illustrate the theoretical results by successfully separating orientations within a Markov equivalence class, which are indistinguishable by conditional independence alone.