Naive Probability: Model-Based Estimates of Unique Events

Abstract We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P(A), P(B), P(A and B), disjunctions: P(A), P(B), P(A or B or both), and conditional probabilities P(A), P(B), P(B|A). They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning

A mental model theory of set membership

Assertions of set membership, such as Amy is an artist, should not be confused with those of set inclusion, such as All artists are bohemians. Membership is not a transitive relation, whereas inclusion is. Cognitive scientists have neglected the topic, and so we developed a theory of inferences yielding conclusions about membership, e.g., Amy is a bohemian, and about non-membership, Abbie is not an artist. The theory is implemented in a computer program, mReasoner, and it is based on mental models. The theory predicts that inferences that depend on a search for alternative models should be more difficult than those that do not. An experiment corroborated this prediction. The program contains a parameter, σ, which determines the probability of searching for alternative models. A search showed that its optimal value of.58 yielded a simulation that matched the participant’s accuracy in making inferences. We discuss the results as a step towards a unified theory of reasoning about sets

What makes intensional estimates of probabilities inconsistent?

Individuals are happy to make estimates of the probabilities of unique events. Such estimates have no right or wrong answers, but when they suffice to determine the joint probability distribution, they should at least be consistent, yielding one that sums to unity. Mental model theory predicts two main sources of inconsistency: the need to estimate the probabilities that events do not happen, and the need to estimate conditional probabilities as opposed, say, to conjunctive probabilities. Experiments 1 and 2 corroborated the first prediction: when the number of estimates of non-events increased for a problem, so did the degree of overall inconsistency. Experiment 3 corroborated the second prediction: when the number of estimates of conditional probabilities increased, the degree of overall inconsistency was larger as well

The probabilities of unique events

Many theorists argue that the probabilities of unique events, even real possibilities such as President Obama’s re-election, are meaningless. As a consequence, psychologists have seldom investigated them. We propose a new theory (implemented in a computer program) in which such estimates depend on an intuitive non-numerical system capable only of simple procedures, and a deliberative system that maps intuitions into numbers. The theory predicts that estimates of the probabilities of conjunctions should often tend to split the difference between the probabilities of the two conjuncts. We report two experiments showing that individuals commit such violations of the probability calculus, and corroborating other predictions of the theory, e.g., individuals err in the same way even when they make non-numerical verbal estimates, such as that an event is highly improbable

3D scatterplots of estimates of P(<i>A</i>), P(<i>B</i>), and P(<i>A&B</i>) and 2D scatterplots of estimates of P(<i>A</i>) and P(<i>B</i>) in Experiment 1.

<p>Panel A shows the estimates of 2000 simulated runs of the computational model and its best fitting linear regression plane, and Panel B shows participants' estimates. Participants' estimates were separated by whether the estimate reflected zero, one, or two violations of the JPD. A violation was defined as a negative probability in the JPD extrapolated from the estimates. In the 2D scatterplots, estimates of P(<i>A&B</i>) correspond to the size of points such that larger points indicate larger estimates.</p

The two different orders of estimates in Experiment 1, the percentage of participants' violations of the JPD, and the latencies (in s) of participants' estimates of the three different probabilities.

<p>The two different orders of estimates in Experiment 1, the percentage of participants' violations of the JPD, and the latencies (in s) of participants' estimates of the three different probabilities.</p

3D scatterplots of estimates of P(<i>A</i>), P(<i>B</i>), and P(<i>A&B</i>) and 2D scatterplots of verbal scale (Panel A) and numerical scale (Panel B) estimates of P(<i>A</i>) and P(<i>B</i>) in Experiment 2 (see <b>Figure 1</b> for an explanation of zero, one, or two violations).

<p>Participants' estimates were separated by whether the estimate reflected zero, one, or two violations of the JPD. A violation was defined as a negative probability in the JPD extrapolated from the estimates. In the 2D scatterplots, estimates of P(<i>A&B</i>) correspond to the size of points such that larger points indicate larger estimates.</p

The conjunctive events of the 16 contents for the problems in Experiment 1, and their respective mean percentage probability estimates.

<p>The table presents the contents in which <i>A</i> decreased the likelihood of <i>B</i>, and then the contents in which <i>A</i> increased the likelihood of <i>B</i>.</p