Sufficient Causes: On Oxygen, Matches, and Fires

1 Introduction

This note illustrates the use of structural models in counterfactual reasoning. In particular, it demonstrates the computation of a quantity denoted PS – the probability of sufficiency, which plays an important role in commonsense reasoning, as well as in legal and medical applications [1], [2], [3], [4].

To motivate the analysis, we will use the classical example of Oxygen, Matches, and Fire which The Book of Why [5, p. 289] describes as follows:

“A fire broke out after someone struck a match, and the question is ‘What caused the fire, striking the match or the presence of oxygen in the room?’ Note that both factors are equally necessary, since the fire would not have occurred absent one of them. So, from a purely logical point of view, the two factors are equally responsible for the fire. Why, then, do we consider lighting the match a more reasonable explanation of the fire than the presence of oxygen?”

The intuitive explanation invokes the notion of prevalence, or anticipation:

“The person who lit the match ought to have anticipated the presence of oxygen, whereas nobody is generally expected to pump all the oxygen out of the house in anticipation of a match-striking ceremony.”

This intuition can also be captured by the notion of sufficiency; striking a match is more likely to be sufficient for the fire than the presence of oxygen. The language of counterfactuals permits us to make this distinction precise as follows: For any two variables, X and Y, define a quantity PS as “the probability that event X=1 would be sufficient to producing outcome Y=1.” Using parenthetical counterfactual notation Y(X=1) to denote “the value that Y would attain had X been 1,” PS can be written as [2, Definition 9.2.1]:

In words, PS asks us to imagine a situation where X=0 and Y=0 and to test how likely it is for Y to turn into Y=1 if X were to change (counterfactually) from X=0 to X=1. Eq. (1) thus quantifies the capacity of X to produce an outcome Y=1 in situations where the outcome is absent. The reason that we must quantify this hypothetical event with probabilities is that both X and Y are random variables, subjected to the whims of unknown factors, some creating situations in which X produces Y, and some creating other situations where X does not produces Y. Eq. (1) quantifies production over all situations, weighted by their likelihood.

We will now compute PS for each of Oxygen and Match and compare their magnitudes. We start by specifying a structural causal model (SCM) for the variables

assuming that F responds to M and OX through the logical ‘AND’ function

Additionally, we assume that, prior to observing the fire, the probabilities for M and OX were:

with pox≫pm, since match-lighting is a rare event and the presence of oxygen is common.

We are now set to derive the probability of sufficiency (Eq. (1)) for both Oxygen and Match using a three-step procedure developed in [2, p. 206]. But before presenting this derivation, it is important that we step back and understand the significance of this exercise. Note that we are about to derive a counterfactual expression, Eq. (1), from a model that is totally void of such expressions. Instead, the model depicts the science behind the fire story in the form of a Boolean function, Eq. (2), and two probabilities, Eq. (3), that can be estimated from the data. This stands in sharp contrast to conventional methods of estimating counterfactual quantities in the potential outcome framework which, invariably, start with counterfactual assumptions justified by drawing analogies to treatments assignments, or “well-defined” manipulations in controlled randomized experiments [6], [7], [8], [9], [10], [11].

There is nothing resembling treatments or experimental manipulation in the function of Eq. (2). One can, of course envision a variety of experiments on the process described in Eq. (2) but those would be conducted to interrogate the process, not to define it. The process itself is specified independently of any envisioned manipulations. See [5, pp. 144–150] for discussion of how experiments interrogate Nature, rather than define it.

This difference between causal models and manipulation-based models is essential for understanding the significance of the exercise described in this note. We will assess counterfactual quantities (Eq. (1)) directly from Nature (Eq. (2)) without asking an investigator to translate Nature into a set of counterfactual statements, prior to commencing the analysis. This we now demonstrate using the three-step procedure derived in [2, p. 206]. 1. Abduction, 2. Action, and 3. Prediction.

Metaphorically, these steps call for: 1. Updating history in light of the available evidence, 2. Bending the course of history (minimally) to comply with the antecedent, and 3. Predicting the outcome based on the updated past and modified model.

2 Formal derivation

Problem: Compute PS(M) and PS(OX), where (from (1)):

Assumptions:

The model is given by the graph G below, where U1 and U2 represent unobserved factors which affect OX and M, respectively. For simplicity, we will assume these factors to be independent, as shown in Fig. 1.

Figure 1

A graph G representing the structural model of Eq. (2) driven by two unobserved factors, U1 and U2.

We shall now derive PS(OX) and PS(M) by applying the three-step algorithm to the model of Fig. 1.

1. Abduction: We need to update the prior probabilities pox and pm in light of the evidence F=0. This amounts to computing pox′ and pm′ for model G, in which F is known to be False (F=0) (the situation prior to observing fire, as in Fig. 2.

   Derivation:

   pox′=P(OX=1|F=0)=P(OX=1,F=0)/P(F=0)==P(OX=1,M=0)/1−P(OX=1,M=1)=pox(1−pm)/(1−poxpm)pm′=P(M=1|F=0)=P(M=1,F=0)/P(F=0)==P(M=1,OX=0)/1−P(OX=1,M=1)=pm(1−pox)/(1−poxpm)

   For pm≪1 and pox≈1 we obtain:

   (4)pox′≈poxandpm′≈pm

   

   Figure 2

   A graph G′ representing the model of Eq. (2) in the absence of fire.

   The reason is clear; the updated priors are simply the old priors re-normalized, after excluding the event F=1, which is very rare. An identical result holds therefore when U1 and U2 are dependent (see Appendix A).

2. Action: To compute PS(M), we take the updated model of Fig. 2 and simulate the action do(M=1). This results in the graph GM=1, of Fig. 3:

   

   Figure 3

   A graph GM=1 representing the simulated action do(M=1) on the updated model of Fig. 2, yielding P(F=1)=pox′.

   Similarly, to compute PS(OX), we simulate the action do(OX=1), leading to graph GOX=1, of Fig. 4.

   

   Figure 4

   A graph GOX=1 representing the simulated action do(OX=1) on the updated model of Fig. 2, yielding P(F=1)=pm′.

3. Prediction: To complete the derivation of PS(M), we now compute P(F=1) in GM=1, yielding:

   PS(M)=P(F=1)inGM=1=pox′≈pox

   Likewise, to compute PS(OX), we compute P(F=1) in GOX=1 giving:

   PS(OX)=P(F=1)inGOX=1=pm′≈pm

   Thus, we have

   (5)PS(M)≈poxandPS(OX)≈pm

   and PS(M)≫PS(OX) as expected.

3 Conclusions and related works

The primary purposes of this note have been: (1) To demonstrate that counterfactuals are derivable algorithmically from common scientific knowledge, and are not needed as inputs for causal analysis. (2) To empower researchers with methods of estimating counterfactuals directly from functional description of their problems. We have demonstrated these two capabilities by computing PS, the probability of sufficiency, in the context of the classical Oxygen-Match-Fire example, which is pivotal for understanding causal explanations. Using this computation we obtained a formal confirmation of the intuition that lighting the match is the more plausible cause of the fire, not the presence of oxygen.

A brief historical overview of this problem and previous works towards its solution should help the reader appreciate its context and importance.

The most common conception of causation – that the effect E would not have occurred in the absence of the cause C – goes back to Hume (1748) [12], and captures the notion of “necessary causation.” The probabilistic version of necessary causation (PN) is behind many judicial standards. In tort law, for example, damage should be paid if and only if it is more probable than not that damage would not have occurred but for the defendant action.

But causation has two faces, necessary and sufficient. The distinction between the two was first articulated by John Stuart Mill (1843) [13], and has received semi-formal explications in the 1960s, first using conditional probabilities [14] and then using logical implications [15]. Both explications suffer from basic semantical difficulties, since probabilities and classical logic are too crude to capture the logic of counterfactual conditionals ([16]; [2, pp. 249–256, 313–316]). The popular “Sufficient Component” model of Kenneth Rothman [17] is essentially equivalent to Mackie’s “INUS condition” and inherits the semantical difficulties noted in [16]. Nevertheless, the graphical schematics of Rothman’s “causal pies” were found very effective in teaching epidemiologists how to represent interacting causes as Boolean functions in disjunctive form. Additionally, counterfactual interpretations of Rothman’s model (VanderWeele and Hernán [18] have resolved some of its semantical difficulties. In particular, these interpretations restrict variables from entering the sufficient cause model unless they are parents of the outcome variable in the causal diagram, as depicted in Fig. 1.

Robins and Greenland [3] gave a counterfactual definition for the probability of necessary causation taking counterfactuals as primitives, and assuming that one is in possession of a joint probability function over counterfactual events. Pearl [19] gave definitions for the probabilities of necessary or sufficient causation (or both) based on structural model semantics which, as we have seen in this note leads to effective procedures for computing counterfactuals from a given causal theory [20], [21]. Additionally, this semantics can be characterized by a complete set of axioms [22], [23], which can be used as inference rules in the analysis.

Pearl [19] and Tian and Pearl [24] have derived tight bounds on PS and PN when both observational and experimental data are available. A tool kit for solving counterfactual parameters is given in [25, pp. 116–126].

Our derivation of PS also bears on a recent debate concerning the role of non-manipulable variables in causal inference, specifically, whether variables such as sex or race can be considered “causes” [26], [27]. In our example, oxygen is practically non-manipulable, and yet, the structural model of Fig. 1 treats oxygen and match on equal footing, with oxygen serving as an enabler of fire (see Appendix B). The model further allows for the estimation of the counterfactuals PS(OX) and PS(M) by the same three-step procedure, regardless of how manipulable they are. Such counterfactuals are considered “not well-defined” in the orthodox school of potential outcome, an untenable stance that would prohibit our question “what caused the fire” from being asked, let alone being answered.