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Estimating Causal Effects in Partially Directed Parametric Causal Factor Graphs

Malte Luttermann, Tanya Braun, Ralf Möller, Marcel Gehrke

TL;DR

It is shown how lifting can be applied to causal inference in partially directed graphs, i.e., graphs that contain both directed and undirected edges to represent causal relationships between random variables.

Abstract

Lifting uses a representative of indistinguishable individuals to exploit symmetries in probabilistic relational models, denoted as parametric factor graphs, to speed up inference while maintaining exact answers. In this paper, we show how lifting can be applied to causal inference in partially directed graphs, i.e., graphs that contain both directed and undirected edges to represent causal relationships between random variables. We present partially directed parametric causal factor graphs (PPCFGs) as a generalisation of previously introduced parametric causal factor graphs, which require a fully directed graph. We further show how causal inference can be performed on a lifted level in PPCFGs, thereby extending the applicability of lifted causal inference to a broader range of models requiring less prior knowledge about causal relationships.

Estimating Causal Effects in Partially Directed Parametric Causal Factor Graphs

TL;DR

It is shown how lifting can be applied to causal inference in partially directed graphs, i.e., graphs that contain both directed and undirected edges to represent causal relationships between random variables.

Abstract

Lifting uses a representative of indistinguishable individuals to exploit symmetries in probabilistic relational models, denoted as parametric factor graphs, to speed up inference while maintaining exact answers. In this paper, we show how lifting can be applied to causal inference in partially directed graphs, i.e., graphs that contain both directed and undirected edges to represent causal relationships between random variables. We present partially directed parametric causal factor graphs (PPCFGs) as a generalisation of previously introduced parametric causal factor graphs, which require a fully directed graph. We further show how causal inference can be performed on a lifted level in PPCFGs, thereby extending the applicability of lifted causal inference to a broader range of models requiring less prior knowledge about causal relationships.

Paper Structure

This paper contains 9 sections, 3 theorems, 3 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Previous Work.
  3. Our Contributions.
  4. Structure of this Paper.
  5. Preliminaries
  6. Partially Directed Parametric Causal Factor Graphs
  7. Conditional Independence in PPCFGs
  8. Efficient Estimation of Causal Effects in PPCFGs
  9. Conclusion

Key Result

theorem 1

Let $M = (\boldsymbol A \cup \boldsymbol G, \boldsymbol E)$ denote a ppcfg and let $P(Q \mid do(R'_1 = r'_1, \ldots, R'_k = r'_k))$ be an interventional query with $Q \in gr(\boldsymbol A)$, $R'_1 \in gr(\boldsymbol A), \ldots, R'_k \in gr(\boldsymbol A)$, and $\{Q\} \cap \{R'_1, \ldots, R'_k\} =

Figures (3)

  • Figure 1: A ppcfg modelling the interplay of a company's revenue and its employees' competences and salaries (without input-output mappings of pf).
  • Figure 2: A visualisation of the resulting model when grounding the ppcfg $M$ given in \ref{['fig:ppcfg_example']}, where $\mathrm{dom}(E) = \{alice,bob,charlie\}$.
  • Figure 3: The modified ppcfg obtained after splitting the pf in the ppcfg shown in \ref{['fig:ppcfg_example']} to separate $Comp(alice)$ from $Comp(E)$.

Theorems & Definitions (14)

  • definition 1: Parameterised Random Variable
  • definition 2: Parfactor
  • definition 3: Partially Directed Parametric Causal Factor Graph
  • definition 4: Query
  • definition 5: $\boldsymbol d$-separation
  • remark 1
  • remark 2
  • definition 6: Intervention
  • theorem 1
  • proof
  • ...and 4 more