## Learning Quadratic Variance Function (QVF) DAG Models via OverDispersion Scoring (ODS)

*Gunwoong Park, Garvesh Raskutti*; 18(224):1−44, 2018.

### Abstract

Learning DAG or Bayesian network models is an important problem
in multi-variate causal inference. However, a number of
challenges arises in learning large-scale DAG models including
model identifiability and computational complexity since the
space of directed graphs is huge. In this paper, we address
these issues in a number of steps for a broad class of DAG
models where the noise or variance is signal-dependent. Firstly
we introduce a new class of identifiable DAG models, where each
node has a distribution where the variance is a quadratic
function of the mean (QVF DAG models). Our QVF DAG models
include many interesting classes of distributions such as
Poisson, Binomial, Geometric, Exponential, Gamma and many other
distributions in which the noise variance depends on the mean.
We prove that this class of QVF DAG models is identifiable, and
introduce a new algorithm, the OverDispersion Scoring (ODS)
algorithm, for learning large-scale QVF DAG models. Our
algorithm is based on firstly learning the moralized or
undirected graphical model representation of the DAG to reduce
the DAG search-space, and then exploiting the quadratic variance
property to learn the ordering. We show through theoretical
results and simulations that our algorithm is statistically
consistent in the high-dimensional $p>n$ setting provided that
the degree of the moralized graph is bounded and performs well
compared to state-of-the-art DAG-learning algorithms. We also
demonstrate through a real data example involving multi-variate
count data, that our ODS algorithm is well-suited to estimating
DAG models for count data in comparison to other methods used
for discrete data.

[abs][pdf][bib]