Reasoning strategies for diagnostic probability estimates in causal contexts: Preference for defeasible deduction over abduction?

Abstract

Recently, Meder, Mayrhofer, and Waldmann [1,2] have proposed a model of causal diagnostic reasoning that predicts an interference of the predictive probability, Pr(Effect | Cause), in estimating the diagnostic probability, Pr(Cause | Effect), specifically, that the interference leads to an underestimation bias of the diagnostic probability. The objective of the experiment reported in the present paper was twofold. A first aim was to test the existence of the underestimation bias in individuals. Our results indicate the presence of an underestimation of the diagnostic probability that depends on the value of the predictive probability. Secondly, we investigated whether this bias was related to the type of estimation strategy followed by participants. We tested two main strategies: abductive inference and defeasible deduction. Our results reveal that the underestimation of diagnostic probability is more pronounced under abductive inference than under defeasible deduction. Our data also suggest that defeasible deduction is for individuals the most natural reasoning strategy to estimate Pr(Cause | Effect).

Keywords

• diagnostic inference
• defeasible deduction
• abduction
• causal Bayes nets

-

? All supplementary information as well as all materials, data and scripts for the statistical analyses can be downloaded from: https://osf.io/7yc92/?view_only=57085866d97d48a4bbd75938d3b4a6af. estimates the probability of Cause given Effect, Pr(c | e). In so-called predictive reasoning, causal inferences go from Cause to Effect; here the aim is to estimate the probability of Effect given Cause, Pr(e | c). The present paper focuses on the former type of reasoning. Diagnostic reasoning is emblematic not only in the field of medicine, but also in our everyday lives. For example, we reason from effects to causes when we try to understand why our car refuses to start or why we failed the final year exam. Here, we focus on the most elementary type of diagnostic inference, which involves a single cause–effect relation between two binary events (i.e., events that either occur or do not occur).

In the experiment to be reported, we show that individuals’ reasoning corroborates the predictions of the Structure Induction Model (SIM) recently proposed by Meder, Mayrhofer, and Waldmann [ 1,2 ]. Whereas it is usually assumed that diagnostic judgments should merely be a function of the empirical conditional probability Pr(c | e), the SIM predicts that diagnostic inferences are also systematically affected by the empirical predictive probability Pr(e | c).

We generalize this result by showing that the influence of the predictive probability in the estimation of the diagnostic probability is effective whatever the reasoning strategy followed by the participants. Two strategies have particularly caught our attention: the estimate of Pr(c | e) by abduction, on the one hand, and the estimate through defeasible deduction, on the other hand (see Section 3). 2

Estimate of Pr(c | e) via Causal Bayes Nets Causal Bayes Nets (CBNs) are today the dominant type of model for formalizing causal inferences [ 3,4,5,6,7 ]. Applied to basic diagnostic inferences, CBNs can define basic causal structures with three parameters.

We first find Pc, which represents the cause base rate, considered here as the prior probability of the cause. Knowledge of this parameter generally comes from an external source, for instance, it could be reported by an expert. We then find the causal power of the target cause Wc. This is an unobservable parameter that can only be estimated from the data provided by nature.1 Finally, to be fully characterized, the network must be complemented by a nuisance parameter that represents all possible alternative causes. Associated with the nuisance parameter is Wa, which is analogues to Wc. It corresponds to the aggregate formed from the causal power and the base rates of all possible alternative causes, and it represents the probability that the effect is present while the target cause is absent.

The activation function (noisy-OR type) of the causal network implies that the effect can be generated independently by the target cause, or by the amalgam of the alternative causes, or by these two variables simultaneously [ 3,8,10 ]. In 1 According to power PC theory [ 8 ], causal power represents the probability of a cause, acting alone, to produce a effect: Wc = Pr(e | c)−Pr(e |¬c) / 1−Pr(e |¬c) . Given that causes are never observed alone, this is a theoretical measure that can only be estimated from data. this context, Pr(c | e) is a function of the three parameters defined above, as follows:

Pr(c | e) = 1 − (1 − Pc) × Pc · Wc + WaW−a Pc · Wc · Wa . (1) 2.1

The Structure Induction Model

Meder et al.’s previously mentioned SIM builds on the CBN literature. In their model, the diagnostic probability depends on the parameters Pc, Wc, Wa, but also on the uncertainty concerning the causal structure itself. An individual in a diagnosis situation is often uncertain about which world she inhabits. Figure 1 illustrates the situation in which she does not know whether she is in a world where there is a causal connection between the cause of interest and the effect (structure S1) or whether she is in a world where such a link does not exist (structure S0). In such a situation, estimating the diagnostic probability follows a three-step process: 1. Based on available data D, the first step consists in estimating, by Bayesian inference, the said parameters for each structure, S0 and S1, separately. We will call θ the set of parameters to be estimated. For S0, there will be a guess of Pc and Wa (the value of Wc is set to 0). For the S1 structure, there will be a guess of Pc, Wc, and Wa. The posterior probability distribution of the parameters for a structure Si is given by

Pr(θ | D ; Si) =

Pr(D | θ ; Si) Pr(θ | Si) ,

Pr(D | Si) where the prior probabilities of the parameters, Pr(θ | Si), are assumed to follow a Beta(1, 1) distribution. Under a noisy-OR hypothesis, the likelihood of the data given the parameters values for a structure Si, Pr(D | θ ; Si), are given by Equations 3 (structure S0) and 4 (structure S1):2

Pr(D | θ ; S0) = [(1 − Pc)(1 − Wa)]N(¬c, ¬e) · [(1 − Pc)Wa]N(¬c, e) · [Pc(1 − Wa)]N(c, ¬e) · [Pc Wa]N(c, e) ; (3) (2) (5) (6)

Pr(Si) is the prior probability of structure Si and it was set to 0.5 for both structures. Pr(D | Si) is the likelihood of the data given the structure and is computed by integrating over the likelihood functions of the parameters (see Equations 3 and 4) under structure Si

Pr(D | Si) =

Z Z Z

Pr(D | θ ; Si) Pr(θ | Si) dθ, (7) 2 In these two equations, the terms c and e in the exponent N (c, e), which may also occur negated, denote contingencies in the data about the target cause and the effect. For example, N (¬c, e) denotes the number of cases where the cause is missing and the effect present. 3 A weak empirical contingency between the Cause of interest and the Effect will suggest that Pr(S0 | D) > Pr(S1 | D). A strong contingency between Cause and Effect will instead suggest that Pr(S0 | D) < Pr(S1 | D).

Pr(D | θ ; S1) = [(1 − Pc)(1 − Wa)]N(¬c, ¬e) · [(1 − Pc)Wa]N(¬c, e) · [Pc(1 − Wc)(1 − Wa)]N(c, ¬e) · [Pc(Wc + Wa − Wc.Wa)]N(c, e). (4) 2. The next step consists in estimating the probabilities of the causal structures Si themselves. The calculation of these probabilities is carried out for each of the structures S0 and S1 separately, on the basis of the available empirical data D and the parameters estimated in the previous step. This leads to the posterior probabilities of each structure under the data: Pr(S0 | D) and Pr(S1 | D).3 These two conditional probabilities are given by the probability of the data being given by

Pr(Si | D) =

Pr(D | Si) Pr(Si) ,

Pr(D)

X where Pr(θ | Si) denotes the joint prior probability over the structures’ parameters. For both structures, this probability is assume to follow a Beta(1, 1) distribution. 3. Finally for S0 and S1, from the structure’ parameters and from the posterior probabilities of the structures, one calculates two diagnostic probabilities. These probabilities are computed by integrating over the parameters’ values weighted by their posterior probabilities:

Pr(c | e ; D, Si) =

Pr(c | e ; θ, Si) Z Z Z

Pr(D | θ ; Si) Pr(θ | Si) dθ,

Pr(D | Si) with and

Pr(c | e ; θ, S0) = Pc, Pr(c | e ; θ, S1) =

(Wc + Wa − WcWa)Pc (Wc + Wa − WcWa)Pc + (1 − Pc)Wa .

To obtain a single diagnostic probability Pr(c | e ; D), the diagnostic probabilities for the two structures (see Equation 8) are weighted by the posterior probabilities of the corresponding structure (see Equations 5 and 6) and summed together: (8) (9) (10)

X This posterior probability will thus take into account both the uncertainty concerning the parameters Pc, Wc, Wa, as well as the uncertainty related to the causal structure.4

In the process that leads to the estimation of the ultimate diagnostic probability Pr(c | e ; D), the second step is crucial because it is the main source of the predictions of the SIM (the influence on the diagnostic probability Pr(c | e ; D) of the predictive probability Pr(e | c)). This step consists in determining the probability of each of the structures, knowing the data: Pr(Si | D), for i ∈ {0, 1}. Since individuals have limited cognitive capacities [ 9,11 ] and do not have direct access to the parameters of the structure (e.g., the causal power of the cause of interest Wc is an unobservable parameter), they estimate Pr(Si | D) by examining the contingencies between the Cause of interest and the Effect in the available data. This examination will lead them to estimate the predictive probability Pr(e | c). The calculation of this value functions as a kind of heuristic to approximate Pr(Si | D) [ 12 ].

Thus, when Pr(e | c) is small, this suggests the absence of a link between Cause and Effect. In this case, Pr(S0 | D) will be more salient than Pr(S1 | D). Conversely, when Pr(e | c) is high, it will suggest the existence of a causal link and Pr(S1 | D) will be more salient than Pr(S0 | D). This mechanism constitutes 4 See [2, p. 299] for more formal details involved in these three calculation steps. the main novelty of the SIM, and it predicts the influence of the predictive probability on diagnostic judgments via the evaluation of the probability of the causal structure. 2.2

Key Predictions of the Structure Induction Model

The dependence of the diagnostic probability on the predictive probability is expected to introduce a bias in the estimation of the former. Specifically, the prediction is that when the predictive probability Pr(e | c) is low, individuals will tend to underestimate the diagnostic probability Pr(c | e) to a much greater extent than when Pr(e | c) is high.

This phenomenon can be explained intuitively by considering the situation in which an agent does not know which of the worlds S0 and S1 she inhabits. If the empirical predictive probability Pr(e | c) is small, this suggests that there may not be any link between Cause and Effect, in which case the agent will tend to believe that she is in world S0. She will then be reluctant to attribute diagnostic virtues to the Effect, which in turn will lead her to underestimate the value of the diagnostic probability calculated from the data. On the other hand, if Pr(e | c) is high, the agent will be inclined to believe that there is a causal relationship between Cause and Effect, whence she is likely to conclude that she is in world S1. In that case, the estimate of Pr(c | e) will more objectively reflect the empirical diagnostic probability of the data. This assumption has been confirmed empirically by Meder et al. [ 2 ]. In this study, we want to reproduce and generalize their important finding by testing the evolution of the bias as a function of the strategy that individuals use to estimate the diagnostic probability. 3

Diagnostic Reasoning Strategies: Abduction versus Defeasible Deduction The SIM specifies a rational computation procedure which links the diagnostic judgments of two types of uncertainty (uncertainty about the parameters and uncertainty concerning the causal structure). In the terminology of Marr [ 13 ], this procedure is at the computational level of the cognitive system. However, Meder et al. also showed that the execution of the rational calculation will have consequences at the algorithmic (i.e., psychological) level, in particular, for the influence of the predictive probability on diagnostic judgments. At the algorithmic level, however, the psychological mechanisms underlying the estimation of the diagnostic probability itself have not been precisely described by these authors. We aim to fill this lacuna by introducing the concept of estimation strategy of the diagnostic probability.

Bruner, Goodnow, and Austin [14, p. 54] defined the concept of strategy in a very general way as a pattern of decisions in the acquisition, retention, and use of information to achieve specific objectives. Siegler and Jenkins [15, p. 11] clarified this idea further and defined strategies as sets of procedures or possible methods put in place by individuals to accomplish a given cognitive task.

Various results suggest that individuals can estimate the diagnostic probability by following essentially different inferential strategies. In two experiments, Stilgenbauer and Baratgin [ 16 ] showed that individuals preferred to follow an abductive strategy to estimate Pr(c | e) when the causal rule “If cause then P(effect)” was made sufficiently plausible (the operator P has the intended meaning “there is a chance that”). In that kind of situation, participants’ diagnostic inferences followed a probabilistic Affirming the Consequent schema (henceforth ACp). This pattern of inference is also often recognized as the basic pattern of abduction [ 17,18,19 ]: effect If cause then P(effect)

P(cause)

Stilgenbauer and Baratgin [ 16 ] further showed that individuals came to prefer estimating the diagnostic probability via a defeasible deduction type of reasoning when the plausibility of the rule “If cause then P(effect)” decreased. In that kind of case, individuals tended to reason in accordance with defeasible Modus Ponens (henceforth MPd; see [ 20,21,22 ]): effect If effect then P(cause)

P(cause)

There is much evidence supporting the thought that ACp and MPd are the two major strategies used in diagnostic reasoning. For example, Patel and Groen [ 23 ] showed that medical students naturally form rules of the form “If cause then P(effect)” to evaluate the likelihood of a disease (Cause) from a set of symptoms (Effect), that, in other words, their novice participants engaged in abductive reasoning. By contrast, more experienced doctors were found to typically construct “If effect then P(cause)” rules to arrive at a diagnosis. Similarly, in the field of legal reasoning, Prakken and Renooij [ 24 ] showed that it is formally possible to follow a purely abductive strategy on the one hand and a strategy which uses defeasible deduction schemes on the other hand to trace the causes in a legal case. 4

Experimental Study of Diagnostic Reasoning Strategies for Estimating Pr(c | e) 4.1

Predictions

The objective of this research is to test the predictions of the SIM, taking into account the strategies of diagnostic reasoning followed by participants.5 As pre5 This experiment is an extension of Chapter 6 of Jean-Louis Stilgenbauer’s doctoral thesis [ 25 ]. viously explained, the SIM predicts an influence on the diagnostic probability Pr(c | e) of the predictive probability Pr(e | c), a phenomenon that leads in practice to an underestimation bias of the diagnostic probability. We assume that the bias will be more pronounced for abductive estimates than for estimates obtained via defeasible deduction, given that abduction requires to construct a rule of type “If cause then P(effect),” which is related to the predictive probability Pr(e | c).6 In this type of situation, Pr(e | c) should become more salient to participants, and since this value is connected to the underestimation bias predicted by the SIM, we expect to see here the clearest evidence of interference with the diagnostic estimate. 4.2

Participants

There were 114 participants in this experiment. All were French native speakers, studying at IPC Paris (Faculty of Philosophy and Psychology). Of these participants, 27 were male (M = 20.3, sd = 1.5) and 87 female (M = 20.5, sd = 2). We eliminated 17 subjects from the protocol: 8 because they left the experiment too early, and 9 because the latency of their responses was too high (over 3 minutes). 4.3

Materials

To test the impact of predictive probability on diagnostic probability estimates, we created four diagnostic reasoning situations, keeping empirical Pr(c | e) constant at 0.75 while varying empirical Pr(e | c) across the situations; specifically, Pr(e | c) ∈ {0.1, 0.3, 0.6, 0.9}. In each situation, participants were asked to estimate Pr(c | e). All participants were exposed to the four diagnostic reasoning contexts, which were presented in an order randomized per participant (withinsubjects factor). On the basis of the SIM, we predicted that participants would tend to estimate Pr(c | e) below the empirical probability of 0.75, and that their estimates would depend on the value of Pr(e | c).

The reasoning situations were medical cases in which there were imaginary viruses (Cause) that could infect people. An infected person might or might not develop a characteristic symptom (Effect). Each situation was introduced using a population-based stimulus that summarized a set of observations. The example 6 There is a wealth of evidence showing that people tend to interpret the probability of an indicative conditional, Pr(If A, then C), as the conditional probability Pr(C | A); see, e.g., [ 26,27,28 ]. Admittedly, the conditional we are considering is not “If cause then effect,” whose probability will, for most people, equal Pr(e | c), but rather “If cause then P(effect),” whose probability will, by the same token, equal Pr(Pe | c), which is not necessarily equal to Pr(e | c). But, first, we are only claiming that “If cause then P(effect)” makes Pr(e | c) salient, and that it can do by making Pr(Pe | c) salient (given the similarity between the two expressions). Second, although the two conditional probabilities are formally distinct, anyone who has taught a course in modal logic knows that people have a tendency to collapse iterated or even mixed modalities. As a result, many may fail to distinguish between Pr(e | c) and Pr(Pe | c) in the first place [ 29 ]. in Figure 2 shows the stimulus corresponding to the values Pr(e | c) = 0.1 and Pr(c | e) = 0.75. The other three stimuli were defined by the following probability pairs: Pr(e | c) = 0.3 and Pr(c | e) = 0.75; Pr(e | c) = 0.6 and Pr(c | e) = 0.75; and Pr(e | c) = 0.9 and Pr(c | e) = 0.75.

To test the impact of reasoning strategy (ACp/MPd) on estimates of Pr(c | e), we created four non-overlapping groups of participants (between-subjects factor). Each group was encouraged to estimate the diagnostic probability through a particular type of reasoning. In addition, we created a situation of “free” reasoning to determine the spontaneous and natural estimates of the participants. Participants were randomly assigned to one of four groups: 1. In the first group, participants were encouraged to reason in accordance with the abductive schema ACp. To this end, we introduced with the stimuli the following conditional rule: “If cause then P(effect).” The Cause is a fictitious virus and the Effect is its characteristic symptom. For example, with the material shown in Figure 2, the following rule appeared at the top of the picture: If a patient is infected with the Igorusphère, there is a chance that the patient has nausea. 2. In the second group, the participants were encouraged to reason in accordance with the defeasible Modus Ponens schema (MPd). Now, at the top of the picture, there appeared the default rule: “If effect then P(cause).” For the example of Figure 2, the specific instance was: If a patient has nausea, there is a chance that the patient has been infected with the Igorusphère. 3. In the third group, no rules were proposed, leaving participants completely free to estimate Pr(c | e) in whichever way they preferred. For each stimulus, an accompanying sentence merely emphasized the existence of an association between Cause (disease) and Effect (the symptom). In this group, we first introduced the disease and only then the symptom. With Figure 2 appeared the sentence: There is a chance that Igorusphère infection is associated with nausea. 4. The fourth group was like the third—so this was also a free reasoning group— except that the order in which disease and symptom were introduced was reversed. For example, the sentence that now went with the situation depicted in Figure 2 was: There is a chance that nausea is associated with Igorusphère infection. 4.4

Procedure

The experiment was implemented on the SoSci Survey website (https://www. soscisurvey.de/). Participants were recruited via a group email. Once connected to the experiment, they were informed that they were supposed to answer all four questions. It was emphasized that their answers should be spontaneous and provided fairly quickly. Then, participants were asked to provide some demographic information: gender, age, native language, type and level of the university course. After this information had been recorded, the general experimental instructions were presented. In each of the four situations, participants were asked to estimate Pr(c | e), for the relevant c and e. More specifically, in each situation a new patient with the characteristic symptom of the virus displayed in the stimulus was presented to the participants. They were then asked to assess the chances that this new patient had been infected by the virus. Participants were asked to give their responses on a scale from 0 % to 100 %, which appeared beneath the stimulus, with the cursor initially set at the 0 % end of the scale. 4.5

Results

The results are summarized in Figure 3. A repeated measures analysis of variance (ANOVA) was carried out on the estimates of Pr(c | e) recorded in the free reasoning groups 3 and 4. There was no order effect related to the terms virus/symptoms and symptoms/virus: F (1, 46) = 0.083, p > .05. Accordingly, we merged the data from groups 3 and 4 for the remainder of the analysis. Impact of predictive probability. A repeated measures ANOVA revealed a main effect of the within-subjects factor predictive probability. The estimates of diagnostic probability Pr(c | e) were found to depend on the predictive probability value Pr(e | c). The graph shows the Pr(c | e) estimates to increase (and to approximate the empirical diagnostic probability Pr(c | e) = 0.75) as the predictive probability values Pr(e | c) increase. The effect was highly significant: F (3, 276) = 27.27, p < .001.

Impact of estimation strategy. The ANOVA also showed a main effect of the between-subjects factor diagnostic probability estimation strategy. The effect was significant: F (2, 92) = 3.44, p < .05. Multiple comparisons with Bonferroni correction showed that the Pr(c | e) estimates under defeasible deduction did not vary significantly from estimates made in the free reasoning condition (p > .05). However, Pr(c | e) estimates made under abduction did vary significantly from estimates made under defeasible deduction (p < .01) as well as from estimates made under free reasoning (p < 0.05). The graph shows that Pr(c | e) estimates were more strongly underestimated compared to the empirical value Pr(c | e) = 0.75 under abductive strategy than under defeasible deduction and free reasoning.

Interaction between predictive probability and estimation strategy.

The statistical analysis did not reveal any interaction between the predictive probability Pr(e | c) and the diagnostic estimation strategy: F (6, 276) = 0.17, p > .05. 4.6

Discussion

Our results are clear evidence that participants systematically underestimated Pr(c | e), and that this bias was strongly related to the predictive empirical probability Pr(e | c). For small values of Pr(e | c), the underestimation of Pr(c | e) was maximal, and it decreased as the predictive probability increased. This result is important because it strongly supports the key calculation step of the causal structure probability Pr(Si | D) that was expected in light of Meder, Mayrhofer, and Waldmann’s Structure Induction Model (SIM).

Our data also show that the best estimates of Pr(c | e) were given under defeasible Modus Ponens and under free reasoning (there was no significant difference between these two conditions). This result suggests that defeasible deduction is the natural inference mode to estimate the diagnostic probability. In any case, diagnostic estimates made through abduction deviated much more from the empirical value Pr(c | e) = 0.75. This confirms our initial hypothesis: when the salience of predictive probability Pr(e | c) is increased by the introduction of a causal rule of the form “If cause then P(effect),” the underestimation bias predicted by the SIM gets worse.

Jointly, these results shed interesting new light on the reasoning process underlying defeasible Modus Ponens in causal contexts. In our opinion, this reasoning can be considered as an elementary form of inference to the best explanation [ 30,38 ] because the major premise of the MPd schema (If effect then P(cause)) is an explanation-evoking rule or evidential rule which, according to Pearl [ 32 ], suggests the activation and search for explanation. Coupled with the operating principle of the SIM—in particular the computation step of Pr(Si | D)—this inference tends to support the significant role played by explanatory considerations in the process of determining the diagnostic estimate [ 33,34,35 ].

In this work, we only tested a minimal type of explanatory consideration, one corresponding to the predictive probability Pr(e | c). In actuality, however, people may well exploit more complex predictive probability-based measures, such as Popper’s measure [ 36 ] or Good’s [ 37 ].7 Whether this is so, we leave as a topic for future research. 7 According to Popper’s measure, the explanatory power of c in light of e (plus background knowledge) is given by Pr(e | c) − Pr(e) / Pr(e | c) + Pr(e) . According to Good, that power is given by ln Pr(e | c)/ Pr(e) . See [ 33,38 ] for discussion of these and other probabilistic measures of explanatory power. See [ 39,40 ] for discussion of human (heuristic and analytic) cognitive process in causal reasoning.

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