Comprehension and computation in Bayesian problem solving

Abstract

Humans have long been characterized as poor probabilistic reasoners when presented with explicit numerical information. Bayesian word problems provide a well-known example of this, where even highly educated and cognitively skilled individuals fail to adhere to mathematical norms. It is widely agreed that natural frequencies can facilitate Bayesian inferences relative to normalized formats (e.g., probabilities, percentages), both by clarifying logical set-subset relations and by simplifying numerical calculations. Nevertheless, between-study performance on “transparent” Bayesian problems varies widely, and generally remains rather unimpressive. We suggest there has been an over-focus on this representational facilitator (i.e., transparent problem structures) at the expense of the specific logical and numerical processing requirements and the corresponding individual abilities and skills necessary for providing Bayesian-like output given specific verbal and numerical input. We further suggest that understanding this task-individual pair could benefit from considerations from the literature on mathematical cognition, which emphasizes text comprehension and problem solving, along with contributions of online executive working memory, metacognitive regulation, and relevant stored knowledge and skills. We conclude by offering avenues for future research aimed at identifying the stages in problem solving at which correct vs. incorrect reasoners depart, and how individual differences might influence this time point.

The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares

“…In many areas, such as medicine or law, critical decisions can depend on appropriately applying the Bayes’ rule, e.g., a medical diagnosis can depend on the probability of having a disease given a positive test result. Consider, for instance, a Bayesian reasoning situation like the following version of the medical diagnosis test situation without emphasizing a specific disease (Johnson and Tubau, 2015, p. 3):…”

“…Departing from an evolutionary point of view, Gigerenzer and Hoffrage (1995) and Cosmides and Tooby (1996) suggested the presentation of the statistical information in the format of natural frequencies instead of probabilities, since in our environment single-event probabilities are not observable and thus, during evolution, the human mind adapted to process natural frequencies rather than single-event probabilities. Using the information format of natural frequencies, the statistical information in the situation of medical diagnosis test mentioned above (Johnson and Tubau, 2015, p. 3) can be expressed as the following:…”

“…Therefore, Johnson and Tubau (2015, p. 4) outline “the computational simplification afforded by natural frequencies” which is illustrated in following example when calculating the proportion of women having the disease among all women testing positive:…”

“… Visualization of the statistical information for the medical diagnosis test problem in the version of Johnson and Tubau (2015, p. 3) with the tree diagram and the unit square, respectively .…”

Structural mapping in statistical word problems: A relational reasoning approach to Bayesian inference

“…Educated adults are notoriously poor Bayesian reasoners with explicit numerical information (for recent review see Johnson & Tubau, 2015). While presenting statistical information as natural frequencies is the most widely agreed facilitator of Bayesian inferences (Gigerenzer & Hoffrage, 1995), many reasoners still fail to solve these problems.…”