Asymptotic Causal Inference
  We investigate causal inference in the asymptotic regime as the number of
variables approaches infinity using an information-theoretic framework. We
define structural entropy of a causal model in terms of its description
complexity measured by the logarithmic growth rate, measured in bits, of all
directed acyclic graphs (DAGs), parameterized by the edge density d. Structural
entropy yields non-intuitive predictions. If we randomly sample a DAG from the
space of all models, in the range d = (0, 1/8), almost surely the model is a
two-layer DAG! Semantic entropy quantifies the reduction in entropy where edges
are removed by causal intervention. Semantic causal entropy is defined as the
f-divergence between the observational distribution and the interventional
distribution P', where a subset S of edges are intervened on to determine their
causal influence. We compare the decomposability properties of semantic entropy
for different choices of f-divergences, including KL-divergence, squared
Hellinger distance, and total variation distance. We apply our framework to
generalize a recently popular bipartite experimental design for studying causal
inference on large datasets, where interventions are carried out on one set of
variables (e.g., power plants, items in an online store), but outcomes are
measured on a disjoint set of variables (residents near power plants, or
shoppers). We generalize bipartite designs to k-partite designs, and describe
an optimization framework for finding the optimal k-level DAG architecture for
any value of d \in (0, 1/2). As edge density increases, a sequence of phase
transitions occur over disjoint intervals of d, with deeper DAG architectures
emerging for larger values of d. We also give a quantitative bound on the
number of samples needed to reliably test for average causal influence for a
k-partite design.
