Reasoning with Inconsistent Causal Beliefs

Reasoning with Inconsistent Causal Beliefs John V. McDonnell (john.mcdonnell@nyu.edu) 1 Pedro Tsividis (tsividis@mit.edu) 2 Bob Rehder (bob.rehder@nyu.edu) 1 Abstract Causal reasoning is a critical part of everyday cognition. We ask how people reason about causes when faced with inconsistent sources of knowledge. Causal models arise from multiple sources of information regarding their con- stituent parameters. Knowledge sources may be incon- sistent both within parameters (when one source says a variable should appear often and another says it should appear rarely), and between parameters (when depen- dencies among parameters result in an internally incon- sistent causal model). We provide a normative model for resolving both these sources of conflict. An experiment found that our model of belief integration predicted the qualitative pattern of adults causal inferences under un- certainty. Keywords: Causal Learning; Causal Inference; Proba- bilistic Modeling Introduction From deciding on investment strategies to predicting others’ reactions in social situations, we are constantly making probabilistic judgments about causal systems. However, because we receive information from multiple sources, we are often faced with contradictory beliefs. Consider the problem faced by an epidemiologist trying to understand the causes of chronic hypertension in a particular population. She reads a review paper suggest- ing that smoking tobacco causes hypertension in 50% of patients, and that all other potential causes of hyperten- sion can be ruled out. The epidemiologist knows from survey data that 25% of the population of interest are smokers, and (independently) that 25% have hyperten- sion. If she assumes her maximum likelihood estimates are true, she is left with an incoherent causal model: If smoking is the only cause of hypertension, and is effective half the time, then there should be half as many people with hypertension as there are smokers. Arriving at co- herent causal beliefs will require adjustment. Perhaps hypertension isn’t really as prevalent as she thought, or perhaps the smoking always causes hypertension. In this paper, we propose a normative probabilistic model for reasoning with these inconsistencies and explore the im- plications of that model in an experiment in which par- ticipants receive conflicting sources of evidence. A Model We assume that people’s causal inferences are based on causal graphical models (CGMs), such as the one in Fig- ure 1 in which C1 and C2 are believed to be indepen- dent causes of E. For simplicity, throughout the paper New York University, Department of Psychology, 6 Washington Place, New York, NY 10003 USA Massachusetts Institute of Technology, Department of Brain and Cognitive Sciences, 77 Massachussetts Ave., Cam- bridge, MA 02139 USA we will assume that all variables are binary (present or absent) and that all causal links are generative, bringing about their effects through a noisy-or functional form. We allow for the possibility of additional causes of E not shown in the graph by aggregating them into a single background cause that is always present (for a review of graphical models, see Koller & Friedman, 2009). A fully parameterized CGM is sufficient to answer virtually any question one might want to ask about the variables in- volved, including questions of conditional or joint proba- bility, counterfactual reasoning, and predicting the effect of interventions (Pearl, 2000). However, complications arise when one recognizes that CGMs are constructed from many individual beliefs held by the reasoner. Since these beliefs may come from multiple sources that vary in their reliability, it is inevitable that they will some- times contradict one another. We ask: How should one draw causal inferences in light of such inconsistencies? To answer this question, we first note that inconsisten- cies can be either between or across parameters, where parameters represent one’s beliefs about each constituent of the model. For example, a belief about the probability of a cause being present is one constituent; a belief about the strength of a causal relation is another. In the first section we advance a new proposal for representing un- certainty in CGMs and show how it solves the problem of within-parameter conflicts. We then tackle the more challenging problem of between-parameter conflicts. Resolving conflicts within parameters Consider the problem of representing the base rate of variable C1, represented by parameter c 1 in Figure 1. We suppose that beliefs about base rates may come from first-hand observations (observing the prevalence of C1), explicit, instruction (e.g., hearing that C1 is uncom- mon), and prior beliefs (e.g., a tendency to believe that events of this type arise rarely). Because probability is bounded to the range [0, 1], the information from each of these sources can be encoded as a probability density function in that range. If knowledge is represented as a point estimate of the expected value of the variable combined with a confidence in that point estimate, this information can be encoded as a beta distribution. PDFs of beta distributions representing knowledge sources for C1 are shown at the top of Figure 1. The shape of a beta distribution is controlled by two parameters, α and β, constrained to be positive (we will refer to such pa- rameters as knowledge parameters, or k-parameters, to emphasize that they represent participants’ knowledge and to avoid confusion with the constituent parameters).