# Imprecise Bayesian Networks as Causal Models

## Abstract

**:**

## 1. Introduction

## 2. Precise Bayes Nets as Causal Models

Causal Markov Condition (CMC):For any variable X in $\mathcal{G}$, the value of X is independent of its non-descendants, conditional on its parents.

Screening Off:For any variable X in $\mathcal{G}$, if ${\mathcal{PA}}_{X}$ is the set containing all the parents of X and W is a parent of some variable in ${\mathcal{PA}}_{X}$, then X and W are independent, conditional on ${\mathcal{PA}}_{X}$.

Common Cause:For any disjoint sets of variables $\mathcal{X}$ and $\mathcal{Y}$ in $\mathcal{G}$, if $\mathcal{X}$ and $\mathcal{Y}$ are not unconditionally independent, then there exists a pair of variables $X\in \mathcal{X}$ and $Y\in \mathcal{Y}$ such that either one variable is a descendant of the other, or X and Y have at least one shared ancestor ${A}_{X,Y}$ such that X and Y are independent conditional on ${A}_{X,Y}$.

**Proposition**

**1.**

Minimality:For any graph $\mathcal{G}$, there is a subgraph ${\mathcal{G}}^{*}$ that differs from $\mathcal{G}$ solely with respect to a single edge that is included in $\mathcal{G}$ but absent from ${\mathcal{G}}^{*}$. A graph $\mathcal{G}$ satisfies Minimality if and only if no such subgraph ${\mathcal{G}}^{*}$ satisfies the Causal Markov Condition.

Precise Unconditional Independence:X and Y are independent if and only if, for all values x and y, $P(X=x,Y=y)=P(X=x)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}P(Y=y)$.

Precise Conditional Independence:X and Y are independent conditional on Z if and only if, for all values x, y and z, $P(X=x,Y=y|Z=z)=P(X=x|Z=z)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}P(Y=y|Z=z)$.

## 3. Imprecise Bayes Nets: The Basics

## 4. Two Concepts of Independence

#### 4.1. Strong Independence

Strong Independence:X and Y are strongly independent if and only if, for all values x and y, $\mathcal{K}(X=x,Y=y)\subseteq \mathcal{SI}$.

Conditional Strong Independence:X and Y are strongly independent conditional on Z if and only if, for all values x, y, and z, $\mathcal{K}(X=x,Y=y|Z=z)\subseteq \mathcal{CSI}$.

#### 4.2. Epistemic Independence

Epistemic Independence:X and Y are epistemically independent if and only if, for all values x and y, $\mathcal{K}(X=x|Y=y)\subseteq \mathcal{EI}$ and $\mathcal{K}(Y=y|X=x)\subseteq \mathcal{EI}$.

Conditional Epistemic Independence:X and Y are epistemically independent conditional on Z if and only if, for all values x, y, and z, $\mathcal{K}(X=x|Y=y,Z=z)\subseteq \mathcal{CEI}$ and $\mathcal{K}(Y=y|X=x,Z=z)\subseteq \mathcal{CEI}$.

#### 4.3. Distinguishing the Two Independence Concepts

Urn Example:There are three urns, A, B, and C, each of which contain 100 balls. The balls in each urn are either red or white. Urn A contains 50 red balls, 20 white balls, and 30 balls that are either red or white. Urns B and C each contain 30 red balls, 30 white balls, and 40 that are either red or white. We begin by drawing a ball from Urn A. If the ball drawn from Urn A is red, then we draw a second ball from Urn B. If the ball drawn from Urn A is white, then we draw a second ball from Urn C. Once it is decided which urn to draw from, each ball in that urn is equally likely to be drawn. Let F be a variable whose values denote the color of the first ball drawn, and let S be a variable whose values denote the color of the second ball drawn. Let u be the number of red balls in Urn B, and let v be the number of red balls in Urn C.

**Proposition**

**2.**

- $\mathcal{K}(F=red)=[0.5,0.8]$
- $\mathcal{K}(F=white)=[0.2,0.5]$
- $\mathcal{K}(S=red)=[0.3,0.7]$
- $\mathcal{K}(S=white)=[0.3,0.7]$

- $\mathcal{K}(F=red|S=red)\subseteq [0.5,0.8]$
- $\mathcal{K}(F=red|S=white)\subseteq [0.5,0.8]$
- $\mathcal{K}(F=white|S=red)\subseteq [0.2,0.5]$
- $\mathcal{K}(F=white|S=white)\subseteq [0.2,0.5]$
- $\mathcal{K}(S=red|F=red)\subseteq [0.3,0.7]$
- $\mathcal{K}(S=red|F=white)\subseteq [0.3,0.7]$
- $\mathcal{K}(S=white|F=red)\subseteq [0.3,0.7]$
- $\mathcal{K}(S=white|F=white)\subseteq [0.3,0.7]$

- $P(F=red|S=red)\approx 0.79$
- $P(F=red|S=white)\approx 0.51$
- $P(F=white|S=red)\approx 0.21$
- $P(F=white|S=white)\approx 0.49$
- $P(S=red|F=red)=0.69$
- $P(S=red|F=white)=0.37$
- $P(S=white|F=red)=0.31$
- $P(S=white|F=white)=0.63$

## 5. Problems for an Imprecise Version of CMC

Strong CMC:For any variable X in $\mathcal{I}$, the value of X is strongly independent of its non-descendants, conditional on its parents.Epistemic CMC:For any variable X in $\mathcal{I}$, the value of X is epistemically independent of its non-descendants, conditional on its parents.

Strong Minimality:If we remove an edge from the IBN $\mathcal{I}$, the resulting subgraph does not satisfy Strong CMC.Epistemic Minimality:If we remove an edge from the IBN $\mathcal{I}$, the resulting subgraph does not satisfy Epistemic CMC.

Causal Interpretation Condition:An IBN $\mathcal{I}=\langle \mathcal{V},\mathcal{E},\mathcal{K}\rangle $ can be interpreted causally if and only if for every $P\in \mathcal{K}$, the Bayes net $\mathcal{G}=\langle \mathcal{V},\mathcal{E},P\rangle $ satisfies the precise versions of CMC and Minimality.

## 6. Objections and Responses

#### 6.1. Problems with the Causal Interpretation Condition

Weak Causal Interpretation Condition:An IBN $\mathcal{I}=\langle \mathcal{V},\mathcal{E},\mathcal{K}\rangle $ can be interpreted causally if and only if there is some $P\in \mathcal{K}$ such that the Bayes net $\mathcal{G}=\langle \mathcal{V},\mathcal{E},P\rangle $ satisfies the precise versions of CMC and Minimality.

#### 6.2. Eliminating Epistemic Independence

Strong Independence Only:No two variables X and Y in $\mathcal{I}$ are epistemically independent but not strongly independent.

#### 6.3. Metaphysical Objections to Imprecise Causal Modelling

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 1

**Proof.**

**Lemma**

**A1.**

#### Appendix A.2. Proof of Proposition 2

**Proof.**

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Kinney, D.
Imprecise Bayesian Networks as Causal Models. *Information* **2018**, *9*, 211.
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Kinney D.
Imprecise Bayesian Networks as Causal Models. *Information*. 2018; 9(9):211.
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Kinney, David.
2018. "Imprecise Bayesian Networks as Causal Models" *Information* 9, no. 9: 211.
https://doi.org/10.3390/info9090211