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On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

Peter J. F. Lucas, Eleanora Zullo, Fabio Stella

Abstract

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.

On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

Abstract

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.

Paper Structure

This paper contains 17 sections, 3 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivating example
  3. Mathematical preliminaries
  4. Bayesian networks
  5. Structural causal models
  6. Relationship between exogenous and endogenous variables
  7. How many values do we need?
  8. Formulation of the general case
  9. Offers adding arcs an alternative method?
  10. Transforming a BN to a PSCM
  11. Linear algebra notation and concepts
  12. Linear algebra formulation
  13. Towards linear programming
  14. Existence and uniqueness of solutions of the system of linear inequalities
  15. Permutations of solutions
  16. ...and 2 more sections

Key Result

Proposition 1

Let $P(x_v \mid X_{\underline{\textrm{pa}}(v)})$ be the result of a probability assignment, then it holds that:

Figures (3)

  • Figure 1: Two different Bayesian network representations of the same distribution; the top one a regular Bayesian network, whereas the one at the bottom is a Bayesian network representation of a probabilistic structured causal model; (a): priors; (b) posteriors.
  • Figure 2: Two different Bayesian network representations with two conditional distributions and more values for $R$, before and after entering an intervention $T$; (a): priors; (b): posteriors.
  • Figure 3: Two different Bayesian network representations of the same distribution with one extra arc from $T$ to $R$ with (a): priors; (b) posteriors.

Theorems & Definitions (19)

  • Definition 1: Bayesian network (BN)
  • Definition 2: Structural causal model (SCM)
  • Example 1
  • Example 2
  • Example 3
  • Definition 3: Partition of domain
  • Definition 4: Probability assignment
  • Proposition 1
  • proof
  • Example 4
  • ...and 9 more