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Bayesian Networks and Causal Ecumenism

Abstract

Proponents of various causal exclusion arguments claim that for any given event, there is often a unique level of granularity at which that event is caused. Against these causal exclusion arguments, causal ecumenists argue that the same event or phenomenon can be caused at multiple levels of granularity. This paper argues that the Bayesian network approach to representing the causal structure of target systems is consistent with causal ecumenism. Given the ubiquity of Bayesian networks as a tool for representing causal structure in both philosophy of science and science itself, this result speaks in favor of the ecumenical view, and against rival exclusionary accounts. Gebharter’s (Philos Phenomenol Res 95(2):353–375, 2017) argument that the Bayes nets formalism is consistent with causal exclusion is considered and rebutted.

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Notes

  1. 1.

    Thus, I leave to one side versions of the causal exclusion argument in an interventionist setting that rely on proportionality constraints on causation, e.g. List and Menzies (2009) or Hoffmann-Kolss (2014).

  2. 2.

    Many readers will be familiar with the Faithfulness condition for Bayes nets, which is strictly stronger than Minimality. I choose Minimality over Faithfulness as an adequacy condition for the causal interpretation of Bayes nets, since there is a case to be made that Bayes nets that satisfy Minimality but not Faithfulness are accurate representations of some causal systems. For a perspicuous comparison of the two conditions, see Zhang (2012).

  3. 3.

    See Spirtes (2007) and Eberhardt (2016) for rigorous demonstrations of this point.

  4. 4.

    Note that, since we assume that the value of X is \(x_{i}\), the interventional conditional probability \(p(y_{j}|do(x_{i}))\) denotes the probability that \(Y=y_{j}\) on the supposition that X has been set to \(x_{i}\) via an intervention, as opposed to the probability that \(Y=y_{j}\) conditional on the observation that \(X=x_{i}\).

  5. 5.

    Note that this is distinct from the notion of “stability” in causal models discussed in Woodward (2010).

  6. 6.

    I am grateful to an anonymous reviewer for suggesting this example.

  7. 7.

    Here I have translated Stern and Eva’s thesis into my terminology.

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Acknowledgements

I am grateful to Sander Beckers, Jonathan Birch, Luc Bovens, Paul Daniell, Chris Dorst, Christopher Hitchcock, Christian List, Philip Pettit, Katie Steele, David Watson, an audience at the 2018 meeting of the APA Central Division in Chicago, and several anonymous reviewers for feedback on various versions of this paper.

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Appendix

Appendix

Calculations

Calculation of Inequality (4)

For the purpose of these calculations, yes, no, low, medium, and high are shortened to y, n, l, m and h, respectively. We begin with left-hand term of the inequality in (4). Since \(S^{\prime }\notin \{A^{\prime }\}\) and \(A^{\prime }\) is the sole parent of \(S^{\prime }\), (2) and (3) jointly imply:

$$\begin{aligned} p(S^{\prime }=y|do(A^{\prime }=l))=p(S^{\prime }=y|A^{\prime }=l) \end{aligned}$$
(9)

Which can be calculated as follows:

$$\begin{aligned} p(S^{\prime }= & {} y|do(A^{\prime }=l))=\frac{p(A^{\prime }=l,B^{\prime }=h,S^{\prime }=y)+p(A^{\prime }=l,B^{\prime }=l,S^{\prime }=y)}{p(A^{\prime }=l)} \end{aligned}$$
(10)
$$\begin{aligned} p(S^{\prime }= & {} y|do(A^{\prime }=l))=\frac{.042+.267}{p(A^{\prime }=l)} \end{aligned}$$
(11)

We can calculate \(p(A^{\prime }=l)\) as follows:

$$\begin{aligned} p(A^{\prime }=l)= \,& {} p(A^{\prime }=l,B^{\prime }=h,S^{\prime }=y)+p(A^{\prime }=l,B^{\prime }=h,S^{\prime }=n) \nonumber \\&+ p(A^{\prime }=l,B^{\prime }=l,S^{\prime }=y)+p(A^{\prime }=l,B^{\prime }=l,S^{\prime }=n) \end{aligned}$$
(12)
$$\begin{aligned} p(A^{\prime }=l)= & {} .042+.003+.267+.021=.333 \end{aligned}$$
(13)

This yields the result:

$$\begin{aligned} p(S^{\prime }=y|do(A^{\prime }=l))=\frac{.042+.267}{.333}\approx .93 \end{aligned}$$
(14)

Next, we calculate the right-hand term:

$$\begin{aligned} p(S^{\prime }=y)= \,& {} p(A^{\prime }=h,B^{\prime }=h,S^{\prime }=y)+p(A^{\prime }=h,B^{\prime }=l,S^{\prime }=y) \nonumber \\&+ p(A^{\prime }=m,B^{\prime }=h,S^{\prime }=y)+p(A^{\prime }=m,B^{\prime }=l,S^{\prime }=y) \nonumber \\&+ p(A^{\prime }=l,B^{\prime }=h,S^{\prime }=y)+p(A^{\prime }=l,B^{\prime }=l,S^{\prime }=y) \end{aligned}$$
(15)
$$\begin{aligned} p(S^{\prime }=y)= & {} .042 + .003 + .1 + .067 + .042 + .267=.52 \end{aligned}$$
(16)

Calculation of Eq. (5)

As we have already calculated the right-hand term, we focus on the left-hand term. From the law of total probability, we have:

$$\begin{aligned}&p(S^{\prime }= y|do(B^{\prime }=l))=p(S^{\prime }=y,A^{\prime }=l|do(B^{\prime }=l)) \nonumber \\&\quad + \ p(S^{\prime }=y,A^{\prime }=m|do(B^{\prime }=l)) + p(S^{\prime }=y,A^{\prime }=h|do(B^{\prime }=l)) \end{aligned}$$
(17)

Since \(S^{\prime }\notin \{B^{\prime }\}\), \(A^{\prime }\notin \{B^{\prime }\}\), and \(A^{\prime }\) is a parent of \(B^{\prime }\), the equation above together with (2) and (3) jointly imply:

$$\begin{aligned}&p(S^{\prime }=y|do(B^{\prime }=l))=p(S^{\prime }=y|A^{\prime }=l)p(A^{\prime }=l) \nonumber \\&\quad + \ p(S^{\prime }=y|A^{\prime }=m)p(A^{\prime }=m) + p(S^{\prime }=y|A^{\prime }=h)p(A^{\prime }=h) \end{aligned}$$
(18)
$$\begin{aligned}&p(S^{\prime }=y|do(B^{\prime }=l))=p(A^{\prime }=l,S^{\prime }=y) + p(A^{\prime }=m,S^{\prime }=y) + p(A^{\prime }=h,S^{\prime }=y) \end{aligned}$$
(19)

To see this implication clearly, note that since \(S^{\prime }\notin \{B^{\prime }\}\) and \(A^{\prime }\notin \{B^{\prime }\}\), Eq. (2) and (3) imply that \(p(S^{\prime }=y,A^{\prime }=l|do(B^{\prime }=l))=p(S^{\prime }=y|\mathbf {pa}_{S^{\prime }})p(A^{\prime }=l|\mathbf {pa}_{A^{\prime }})\). Since \(A^{\prime }\) is the sole parent of \(S^{\prime }\) and \(A^{\prime }\) has no parents, \(p(S^{\prime }=y|\mathbf {pa}_{S^{\prime }})p(A^{\prime }=l|\mathbf {pa}_{A^{\prime }})=p(S^{\prime }=y|A^{\prime }=l)p(A^{\prime }=l)\). We repeat these steps to obtain the other summands of (18). By the law of total probability, we obtain:

$$\begin{aligned} p(S^{\prime }=y|do(B^{\prime }=l))=p(S^{\prime }=y)\approx .52 \end{aligned}$$
(20)

Calculation of Inequality (6)

We begin with left-hand term of the inequality in (6). Since \(S^{\prime }\notin \{A\}\) and A is the sole parent of \(S^{\prime }\), (2) and (3) jointly imply:

$$\begin{aligned} p(S^{\prime }=y|do(A=l/m))=p(S^{\prime }=y|A=l/m) \end{aligned}$$
(21)

Which can be calculated as follows:

$$\begin{aligned}&p(S^{\prime }=y|do(A=l/m))\nonumber \\&\quad =\frac{p(A=l/m,B^{\prime }=h,S^{\prime }=y)+p(A=l/m,B^{\prime }=l,S^{\prime }=y)}{p(A=l/m)} \end{aligned}$$
(22)
$$\begin{aligned}&p(S^{\prime }=y|do(A=l/m))=\frac{.142+.334}{p(A=l/m)} \end{aligned}$$
(23)

We can calculate \(p(A^{\prime }=l/m)\) as follows:

$$\begin{aligned} p(A=l/m)= & \,{} p(A=l/m,B^{\prime }=h,S^{\prime }=y)+p(A=l/m,B^{\prime }=h,S^{\prime }=n) \nonumber \\&+ p(A=l/m,B^{\prime }=l,S^{\prime }=y)+p(A=l/m,B^{\prime }=l,S^{\prime }=n) \end{aligned}$$
(24)
$$\begin{aligned} p(A=l/m)= & {} .142+.103+.334 +.088=.666 \end{aligned}$$
(25)

This yields the result:

$$\begin{aligned} p(S^{\prime }=y|do(A=l/m))=\frac{.142+.334}{.666}=.714 \end{aligned}$$
(26)

Finally, we recall the value of the right-hand term \(p(S^{\prime }=y)=.52\).

Proof of Proposition 1

Proof

Assume \(p(y_{j}|do({\mathbf {x}}_{i}))>p(y_{j})\), where \({\mathbf {x}}_{i}\) is a vector of values that contains both \(x_{i}\) and the actual values taken by all parents of Y not on some directed path from X to Y. Let \(\varphi\) be the set of values of Y other than \(y_{j}\). This implies that \(1-p(y_{j}|do({\mathbf {x}}_{i}))<1-p(\varphi )\), which implies in turn that there is a value \(y_{l}\) such that \(p(y_{l}|do({\mathbf {x}}_{i}))<p(y_{l})\). Suppose that the set \(\mathcal {PA}_{X}\) containing all variables that are parents of X and all parents of Y not on the stipulated path from X to Y has the set of possible vectors of values \(\{\mathbf {pa_{X1}},\mathbf {pa_{X2}},\ldots ,\mathbf {pa_{Xq}}\}\). It is well known (see Pearl et al. 2016, p. 59) that we can derive the following:

$$\begin{aligned} p(y_{l}|do({\mathbf {x}}_{i}))=\sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{i},\mathbf {pa_{Xt}})p(\mathbf {pa_{Xt}})<p(y_{l}) \end{aligned}$$
(27)

The law of total probability implies the following, where X has n values:

$$\begin{aligned} p(y_{l})=\sum _{k=1}^{n}\sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{i},\mathbf {pa_{Xt}})p({\mathbf {x}}_{i},\mathbf {pa_{Xt}}) \end{aligned}$$
(28)

This implies that there exists a set of values \({\mathbf {x}}_{k}\) such that \({\mathbf {x}}_{i}\cup {\mathbf {x}}_{k}\setminus {\mathbf {x}}_{i}\cap {\mathbf {x}}_{k}=\{x_{i},x_{k}\}\), and:

$$\begin{aligned} \sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{i},\mathbf {pa_{Xt}})p(\mathbf {pa_{Xt}})<p(y_{l})<\sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{k},\mathbf {pa_{Xt}})p({\mathbf {x}}_{k},\mathbf {pa_{Xt}}) \end{aligned}$$
(29)

The fact that \(p({\mathbf {x}}_{i})>0\) implies that \(p({\mathbf {x}}_{k})<1\), which implies in turn that \(p({\mathbf {x}}_{k},\mathbf {pa_{Xt}})<p(\mathbf {pa_{Xt}})\) for all \(\mathbf {pa_{Xt}}\) with positive probability. It follows from this that:

$$\begin{aligned} p(y_{l})<\sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{k},\mathbf {pa_{Xt}})p({\mathbf {x}}_{k},\mathbf {pa_{Xt}}) < \sum _{t=1}^{q}p(y_{l}|{\mathbf {x}}_{k},\mathbf {pa_{Xt}})p(\mathbf {pa_{Xt}})=p(y_{l}|do({\mathbf {x}}_{k})) \end{aligned}$$
(30)

Which immediately implies that \(p(y_{l}|do(x_{k}))>p(y_{l})\) when we hold fixed all parents of Y not on the stipulated path from X to Y. \(\square\)

Proof of Proposition 2

Proof

Consider a Bayes net \({\mathcal {N}}^{\prime }=\langle {\mathcal {V}}^{\prime },{\mathcal {E}}^{\prime },p(\cdot )\rangle\), with \(C^{\prime }\in {\mathcal {V}}^{\prime }\), where \(C^{\prime }=c^{\prime }_{l}\) causes \(E^{\prime }=e^{\prime }_{s}\). This fact implies that \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))>p(e^{\prime }_{s})\), where \({\mathbf {z}}\) is the actual setting of values for all parents of \(E^{\prime }\) not on some directed path from \(C^{\prime }\) to \(E^{\prime }\). Define an equivalence relation \(\sim\) over the range of \(C^{\prime }\) such that \(c^{\prime }_{l}\sim c^{\prime }_{u}\) if and only if either:

  1. (a)

    \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))>p(e^{\prime }_{s})\) and \(p(e^{\prime }_{s}|do(c^{\prime }_{u},{\mathbf {z}}))>p(e^{\prime }_{s})\), or

  2. (b)

    \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))\le p(e^{\prime }_{s})\) and \(p(e^{\prime }_{s}|do(c^{\prime }_{u},{\mathbf {z}}))\le p(e^{\prime }_{s})\).

Let the range of C be a quotient set of the range of \(C^{\prime }\) according to this equivalence relation. Define a Bayes net \({\mathcal {N}}=\langle {\mathcal {V}},{\mathcal {E}},p(\cdot )\rangle\) such that \({\mathcal {V}}=\{C\}\cup {\mathcal {V}}^{\prime }\setminus \{C^{\prime }\}\) and such that, for any edge \(\langle V^{\prime },C^{\prime }\rangle \in {\mathcal {E}}^{\prime }\) or \(\langle C^{\prime },V^{\prime }\rangle \in {\mathcal {E}}^{\prime }\), there is an edge \(\langle V^{\prime },C\rangle \in {\mathcal {E}}\) or \(\langle C,V^{\prime }\rangle \in {\mathcal {E}}\); all other elements of \({\mathcal {E}}^{\prime }\) are included in \({\mathcal {E}}\). \({\mathcal {N}}\) trivially satisfies (1), (2) and (4); we now show that it satisfies (3) and (5) as well.

Let us begin with condition (3). To show that \({\mathcal {N}}\) satisfies CMC, let \(X^{\prime }\in {\mathcal {V}}^{\prime }\) and \(Y^{\prime }\in {\mathcal {V}}^{\prime }\) be two variables such that \(Y^{\prime }\) is not a descendant of \(X^{\prime }\) in \({\mathcal {N}}^{\prime }\). If \(X^{\prime }\ne C^{\prime }\), \(Y^{\prime }\ne C^{\prime }\), and \(C^{\prime }\notin \mathcal {PA}_{X^{\prime }}\), then the supposition that \({\mathcal {N}}^{\prime }\) satisfies CMC, along with the truth of conditions (1) and (2), implies that \(X^{\prime }\) and \(Y^{\prime }\) are independent, given \(\mathcal {PA}_{X^{\prime }}\), in \({\mathcal {N}}\).

If \(C^{\prime }=X^{\prime }\), then the fact that \({\mathcal {N}}^{\prime }\) satisfies CMC implies that for any values \(c^{\prime }_{l}\), \(y^{\prime }_{o}\), and vector of values \({\mathbf {pa}}_{\mathbf {C}^{\prime }}\) of the variables in \(\mathcal {PA}_{C^{\prime }}\), \(p(c^{\prime }_{l}|y^{\prime }_{o},{\mathbf {pa}}_{\mathbf {C}^{\prime}})=p(c^{\prime }_{l}|{\mathbf {pa}}_{\mathbf {C}^{\prime}})\). If C is a coarsening of \(C^{\prime }\), then for each value \(c_{j}\), each conditional probability \(p(c_{j}|y^{\prime }_{o},\mathbf {pa_{C}})\) and \(p(c_{j}|\mathbf {pa_{C}})\) is a sum of terms of the form \(p(c^{\prime }_{l}|y^{\prime }_{o},{\mathbf {pa}}_{\mathbf {C}^{\prime }})\) and \(p(c^{\prime }_{l}|{\mathbf {pa}}_{\mathbf {C}^{\prime }})\), respectively. Thus, if the latter pair of terms are equal for all triples \((c^{\prime }_{l},y^{\prime }_{o},{\mathbf {pa}}_{\mathbf {C}^{\prime }})\), then the former pair of terms are equal for all values \((c_{j}, y^{\prime }_{o},\mathbf {pa_{C}})\). Thus, C is independent of its non-descendants, given its parents, in \({\mathcal {N}}\).

If \(C^{\prime }=Y^{\prime }\), then the fact that \({\mathcal {N}}^{\prime }\) satisfies CMC implies that for any values \(c^{\prime }_{l}\), \(x^{\prime }_{o}\), and set of values \({\mathbf {pa}}_{\mathbf {X}^{\prime }}\) of the variables in \(\mathcal {PA}_{X^{\prime }}\), \(p(x^{\prime }_{o}|c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})=p(x^{\prime }_{o}|{\mathbf {pa}}_{\mathbf {X}^{\prime }})\). These conditional probabilities can be expressed as the following ratios:

$$\begin{aligned}&p(x^{\prime }_{o}|c^{\prime }_{l},\mathbf {pa_{X^{\prime }}})=\frac{p(x^{\prime }_{o},c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})}{p(c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})} \end{aligned}$$
(31)
$$\begin{aligned}&p(x^{\prime }_{o}|{\mathbf {pa}}_{\mathbf {X}^{\prime }})=\frac{p(x^{\prime }_{o},\mathbf {pa_{X^{\prime }}})}{p({\mathbf {pa}}_{\mathbf {X}^{\prime }})} \end{aligned}$$
(32)

If C is a coarsening of \(C^{\prime }\), then for each value \(c_{j}\), each joint probability \(p(x^{\prime }_{o},c_{j},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\) and \(p(c_{j},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\) is equal to a sum of joint probabilities of the form \(p(x^{\prime }_{o},c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\) and \(p(c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\), respectively. Thus, if \(p(x^{\prime }_{o}|c^{\prime }_{l},{\mathbf {pa}}_{\mathbf {X}^{\prime }})=p(x^{\prime }_{o}|{\mathbf {pa}}_{\mathbf {X}^{\prime }})\) for all triples \((c^{\prime }_{l}, x^{\prime }_{o},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\), then \(p(x^{\prime }_{o}|c_{j},{\mathbf {pa}}_{\mathbf {X}^{\prime }})=p(x^{\prime }_{o}|{\mathbf {pa}}_{\mathbf {X}^{\prime }})\) for all triples \((c_{j},x^{\prime }_{o},{\mathbf {pa}}_{\mathbf {X}^{\prime }})\). Thus, if C is a non-descendant of \(X^{\prime }\) in \({\mathcal {N}}\), then \(X^{\prime }\) is independent of C, given \(X^{\prime }\)’s parents, in \({\mathcal {N}}\). The immediately preceding analysis could be repeated if \(C\in \mathcal {PA}_{X^{\prime }}\), to show that any variable \(X^{\prime }\) is independent of its non-descendants in \({\mathcal {N}}\), given its parents, when those parents include C. Together, these results show that \({\mathcal {N}}\) satisfies CMC. Minimality can be achieved by stipulation, by simply removing any edges that are not necessary for \({\mathcal {N}}\) to satisfy CMC.

Finally, we can show that (5) is true. Suppose that \(C^{\prime }=c^{\prime }_{l}\) implies that \(C=c_{j}\). In other words, \(c^{\prime }_{l}\) is mapped to \(c_{j}\) in the coarsening function from the range of \(C^{\prime }\) to the range of C. We know that \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))>p(e^{\prime }_{s})\), and \(c^{\prime }_{l}\) is \(\sim\)-related to all and only those values of \(C^{\prime }\) such that conditioning on an intervention bringing about those values increases the probability that \(E^{\prime }=e^{\prime }_{s}\), relative to its marginal probability. Thus, the conditional probability \(p(e^{\prime }_{s}|do(c_{j},{\mathbf {z}}))\) is a sum of terms of the form \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))\), each of which is such that \(p(e^{\prime }_{s}|do(c^{\prime }_{l},{\mathbf {z}}))>p(e^{\prime }_{s})\). This implies that \(p(e^{\prime }_{s}|do(c_{j},{\mathbf {z}}))>p(e^{\prime }_{s})\), and therefore that \(C=c_{j}\) causes \(E^{\prime }=e^{\prime }_{s}\). \(\square\)

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Kinney, D. Bayesian Networks and Causal Ecumenism. Erkenn (2020). https://doi.org/10.1007/s10670-020-00343-z

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