# Learning Belief Network Structure From Data under Causal Insufficiency

## TL;DR

This paper investigates the problem of learning belief network structures from data in situations of causal insufficiency, aiming to determine the necessary conditions for accurate structure recovery and the role of hidden variables.

## Contribution

It addresses the open question of whether bidirectional edges are sufficient for belief network recovery under causal insufficiency or if additional hidden variables are needed.

## Key findings

- Identifies conditions under which belief networks can be learned from data with causal insufficiency.
- Clarifies the role of bidirectional edges and hidden variables in structure learning.
- Provides theoretical insights into the limitations of current causal inference algorithms.

## Abstract

Though a belief network (a representation of the joint probability distribution, see [3]) and a causal network (a representation of causal relationships [14]) are intended to mean different things, they are closely related. Both assume an underlying dag (directed acyclic graph) structure of relations among variables and if Markov condition and faithfulness condition [15] are met, then a causal network is in fact a belief network. The difference comes to appearance when we recover belief network and causal network structure from data. A causal network structure may be impossible to recover completely from data as not all directions of causal links may be uniquely determined [15]. Fortunately, if we deal with causally sufficient sets of variables (that is whenever significant influence variables are not omitted from observation), then there exists the possibility to identify the family of belief networks a causal network belongs to [16]. Regrettably, to our knowledge, a similar result is not directly known for causally insufficient sets of variables. Spirtes, Glymour and Scheines developed a CI algorithm to handle this situation, but it leaves some important questions open. The big open question is whether or not the bidirectional edges (that is indications of a common cause) are the only ones necessary to develop a belief network out of the product of CI, or must there be some other hidden variables added (e.g. by guessing). This paper is devoted to settling this question.