A Rational Model of Elemental Diagnostic Inference

A Rational Model of Elemental Diagnostic Inference Bjorn Meder (meder@mpib-berlin.mpg.de) Ralf Mayrhofer (rmayrho@uni-goettingen.de) Michael R. Waldmann (michael.waldmann@bio.uni-goettingen.de) Department of Psychology, University of Gottingen, Gosslerstr. 14, 37073 Gottingen, Germany sulting from a rational inference strategy that is well adapted to the goal of acquiring and using causal knowledge. Abstract Whereas the traditional normative benchmark for diagnostic reasoning from effects to causes is provided by purely statis- tical norms, we here approach the task from the perspective of rational causal inference. The core feature of the presented model is the assumption that diagnostic inferences are con- strained by hypotheses about the causal texture of the domain. As a consequence, the model’s predictions systematically de- viate from classical, purely statistical norms of diagnostic in- ference. In particular, the analysis reveals that diagnostic judgments should not only be influenced by the probability of the cause given the effect, but also be systematically affected by the predictive relation between cause and effect. This pre- diction is tested in three studies. The obtained pattern of diag- nostic reasoning is at variance with the traditional statistical norm but consistent with a model of rational causal inference. “(aive Bayes” as a (orm of Diagnostic Inference Let C denote a binary cause and E a binary effect, and let c + , c − and e + , e − indicate the presence and absence, respectively, of these events. Making a diagnostic judgment from effect to cause can then be expressed as estimating the conditional probability of the cause given the effect, P(c + |e + ). Given a joint frequency distribution over C and E the empirical conditional probability P(c + |e + ) can be directly estimated from the frequency of co-occurrences (·). Alternatively, one can use Bayes’ rule to derive this probability from the conditional probability of the effect given the cause, P(e + |c + ), the base rate of the target cause, P(c + ), and the marginal probability of the effect, P(e + ): Keywords: Rational model; Causal learning; Causal reason- ing; Bayesian inference; Computational Modeling Introduction In this paper we present a rational analysis of diagnostic reasoning – the process of reasoning from effects to causes. Diagnostic inferences are not only ubiquitous in medicine, but also in everyday reasoning. For example, we reason from effects to causes when we try to explain why our car does not start or when we try to identify the causes of why our computer crashed once again. Whereas the traditional normative yardstick for such inferences is provided by pure- ly statistical norms, we use the framework of causal-model theory (e.g., Pearl, 2000; Waldmann & Holoyak, 1992; Waldmann, Hagmayer, & Blaisdell, 2006) and causal Baye- sian inference (Griffiths & Tenenbaum, 2005; Lu, Yuille, Liljeholm, Cheng, & Holyoak, 2008) to elucidate the rele- vant kinds of inputs, computations, and outputs involved in diagnostic reasoning. We here focus on the most basic type of diagnostic infe- rence, which involves a single cause-effect relation between two binary events. Based on a rational analysis of such diagnostic inferences we have developed a computational model that details the influence of competing hypotheses about causal structure and causal strength. Whereas it is usually assumed that diagnostic judgments should merely be a function of the empirical conditional probability P(Cause | Effect), our analysis reveals that diagnostic infe- rences should also be systematically affected by the predic- tive probability P(Effect | Cause) and by the causal power (Cheng, 1997) of the target cause. We tested the model’s predictions in three studies. While the observed pattern of reasoning appears irrational from a purely statistical pers- pective, our analyses suggest that it may be viewed as re- We refer to this approach as naїve Bayes because under this view the application of Bayes’ rule is nothing but an ele- mentary result of standard probability theory. In particular, no reference is made to the generative causal processes underlying the observed events, and no uncertainty about parameter estimates is assumed in these computations. This use of Bayes’ rule provides the classical statistical norm to which peoples’ diagnostic judgments usually have been compared (e.g., Kahneman & Tversky, 1973). Several studies have shown that peoples’ judgments often substan- tially deviate from this norm and have attempted to pinpoint factors which lead people to conform to this norm (e.g., Gigerenzer & Hoffrage, 1995). However, the prescriptive validity of this statistical norm has rarely been questioned (but see Krynski & Tenenbaum, 2007). We suggest that approaching diagnostic inferences from the perspective of causal reasoning may provide a more appropriate standard of rational diagnostic inference and a better descriptive model of peoples’ diagnostic judgments. A Rational Model of Diagnostic Inference The core idea behind our model is the assumption that diag- nostic inferences operate over causal representations that are estimated from data (cf. Krynski & Tenenbaum, 2007). Thus, the data we encounter are typically interpreted as arising from some unobserved causal processes, and our inference goal when making predictive and diagnostic infe- rences is to reason about causal relations, not about the noisy data we perceive.