# 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.**

## References

- Antonucci, A.; Salvetti, A.; Zaffalon, M. Assessing debris flow hazard by credal nets. In Soft Methodology and Random Information Systems; Springer: Berlin/Heidelberg, Germany, 2004; pp. 125–132. [Google Scholar]
- Joyce, J.M. A defense of imprecise credences in inference and decision making. Phil. Perspect.
**2010**, 24, 281–323. [Google Scholar] [CrossRef] - Bradley, S. Imprecise probabilities. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2016. [Google Scholar]
- Pearl, J. Causality; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Spirtes, P.; Glymour, C.N.; Scheines, R. Causation, Prediction, and Search; MIT Press: Cambridge, MA, USA, 2000. [Google Scholar]
- Corani, G.; Antonucci, A.; Zaffalon, M. Bayesian Networks with Imprecise Probabilities: Theory and Application to Classification; Springer: Berlin/Heidelberg, Germany, 2012; pp. 49–93. [Google Scholar]
- Couso, I.; Moral, S.; Walley, P. Examples of independence for imprecise probabilities. In Proceedings of the First Symposium on Imprecise Probabilities and Their Applications (ISIPTA), Ghent, Belgium, 29 June– 2 July 1999. [Google Scholar]
- Cozman, F.G. Sets of probability distributions, independence, and convexity. Synthese
**2012**, 186, 577–600. [Google Scholar] [CrossRef] - Hausman, D.M.; Woodward, J. Independence, invariance and the causal Markov condition. Br. J. Phil. Sci.
**1999**, 50, 521–583. [Google Scholar] [CrossRef] - Woodward, J. Making Things Happen: A Theory of Causal Explanation; Oxford University Press: New York, NY, USA, 2003. [Google Scholar]
- Reichenbach, H. The Direction of Time; Dover Publications: Mineola, NY, USA, 1956. [Google Scholar]
- Verma, T.; Pearl, J. Equivalence and Synthesis of Causal Models. In Proceedings of the Sixth Conference on Uncertainty in Artijicial Intelligence, Cambridge, MA, USA, 27–29 June 1990; pp. 220–227. [Google Scholar]
- Williamson, J. Bayesian Nets and Causality: Philosophical and Computational Foundations; Oxford University Press: New York, NY, USA, 2005. [Google Scholar]
- Wronski, L. Reichenbach’s Paradise: Constructing the Realm of Probabilistic Common “Causes”; de Gruyter: Berlin, Germany, 2014. [Google Scholar]
- Zhang, J. A comparison of three occam’s razors for markovian causal models. Br. J. Phil. Sci.
**2012**, 64, 423–448. [Google Scholar] [CrossRef] - Machamer, P.K.; Darden, L.; Craver, C.F. Thinking about mechanisms. Phil. Sci.
**2000**, 67, 1–25. [Google Scholar] [CrossRef] - Woodward, J. What is a mechanism? A counterfactual account. Phil. Sci.
**2002**, 69, 366–377. [Google Scholar] [CrossRef] - Glennan, S. Rethinking mechanistic explanation. Phil. Sci.
**2002**, 69, 342–353. [Google Scholar] [CrossRef] - Levi, I. On Indeterminate probabilities. J. Phil.
**1974**, 71, 391–418. [Google Scholar] [CrossRef] - Levi, I. The Enterprise of Knowledge: An Essay on Knowledge, Credal Probability, and Chance; MIT Press: Cambridge, MA, USA, 1980. [Google Scholar]
- Walley, P. Statistical Reasoning with Imprecise Probabilities; Chapman and Hall: London, UK, 1991. [Google Scholar]
- Seidenfeld, T.; Schervish, M.J.; Kadane, J.B. Coherent choice functions under uncertainty. Synthese
**2010**, 172, 157–176. [Google Scholar] [CrossRef] - Elkin, L.; Wheeler, G. Resolving peer disagreements through imprecise probabilities. Noûs
**2018**, 52, 260–278. [Google Scholar] [CrossRef] - De Campos, C.P.; Cozman, F.G. Computing lower and upper expectations under epistemic independence. Int. J. Approx. Reason.
**2007**, 44, 244–260. [Google Scholar] [CrossRef] [Green Version] - List, C.; Pivato, M. Emergent chance. Phil. Rev.
**2015**, 124, 119–152. [Google Scholar] [CrossRef] [Green Version]

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kinney, D.
Imprecise Bayesian Networks as Causal Models. *Information* **2018**, *9*, 211.
https://doi.org/10.3390/info9090211

**AMA Style**

Kinney D.
Imprecise Bayesian Networks as Causal Models. *Information*. 2018; 9(9):211.
https://doi.org/10.3390/info9090211

**Chicago/Turabian Style**

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