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(A) Full natural frequency tree for the Green and Mehr (1997) data on 89 patients with severe chest pain. The goal is to determine whether these patients are at high or low risk for heart disease. ST denotes a particular pattern in the electro cardiogram, CP denotes chest pain, OS denotes “at least one other symptom,” “+” denotes present, and “–” denotes absent. Numbers in circles denote number of patients. (B) Fast-and-frugal classification tree obtained by pruning the natural frequency tree. The ranking of cues and the exit structure are determined by the ZigZag method (in the present case, ZigZag-val and ZigZag-sens, as explained in the text, lead to the same trees). Questions in rectangles specify which cues are looked up at this level for each of the patients in the corresponding circles in (A). Depending on whether this cue value is positive or negative, either a new question is asked or the tree in (B) is exited and a classification decision is made (oval). The accuracy of these classification decisions is shown by the number of patients below these oval exit nodes: The number of patients who actually had a heart attack is displayed in the left of the two adjacent end nodes in the lowest layer, and the number of those who did not have one is displayed in the corresponding end node on the right. All patients to the left of the vertical bar in Figure 1B are classified as high risk, and all patients to its right are classified as low risk.
18,601 views
71 citations
Original Research
26 August 2015
Effects of visualizing statistical information – an empirical study on tree diagrams and 2 × 2 tables
Karin Binder
1 more and 
Georg Bruckmaier
Four resulting visualizations of the respective information format (mammography problem).

In their research articles, scholars often use 2 × 2 tables or tree diagrams including natural frequencies in order to illustrate Bayesian reasoning situations to their peers. Interestingly, the effect of these visualizations on participants’ performance has not been tested empirically so far (apart from explicit training studies). In the present article, we report on an empirical study (3 × 2 × 2 design) in which we systematically vary visualization (no visualization vs. 2 × 2 table vs. tree diagram) and information format (probabilities vs. natural frequencies) for two contexts (medical vs. economical context; not a factor of interest). Each of N = 259 participants (students of age 16–18) had to solve two typical Bayesian reasoning tasks (“mammography problem” and “economics problem”). The hypothesis is that 2 × 2 tables and tree diagrams – especially when natural frequencies are included – can foster insight into the notoriously difficult structure of Bayesian reasoning situations. In contrast to many other visualizations (e.g., icon arrays, Euler diagrams), 2 × 2 tables and tree diagrams have the advantage that they can be constructed easily. The implications of our findings for teaching Bayesian reasoning will be discussed.

13,485 views
63 citations
Review
27 July 2015

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.

7,450 views
41 citations
11,116 views
47 citations
10,075 views
49 citations
Stacked bar chart showing probabilities assigned to events and their negations. B, Belief group; I, Inference group; 48, full set of conditionals; 24, reduced set; p, antecedent event (black bar p, white bar not-p); q, consequent event (black bar q, white bar not-q).
Original Research
08 April 2015
Uncertain deduction and conditional reasoning
Jonathan St. B. T. Evans
1 more and 
David E. Over

There has been a paradigm shift in the psychology of deductive reasoning. Many researchers no longer think it is appropriate to ask people to assume premises and decide what necessarily follows, with the results evaluated by binary extensional logic. Most every day and scientific inference is made from more or less confidently held beliefs and not assumptions, and the relevant normative standard is Bayesian probability theory. We argue that the study of “uncertain deduction” should directly ask people to assign probabilities to both premises and conclusions, and report an experiment using this method. We assess this reasoning by two Bayesian metrics: probabilistic validity and coherence according to probability theory. On both measures, participants perform above chance in conditional reasoning, but they do much better when statements are grouped as inferences, rather than evaluated in separate tasks.

9,056 views
71 citations
An illustration of a natural sampling framework: the total population (100) is categorized into groups (5/95) and those groups are categorized into parallel sub-groups below that.
8,385 views
49 citations
Article Cover Image
Original Research
25 February 2015
Bayesian reasoning with ifs and ands and ors
Nicole Cruz
2 more and 
David E. Over

The Bayesian approach to the psychology of reasoning generalizes binary logic, extending the binary concept of consistency to that of coherence, and allowing the study of deductive reasoning from uncertain premises. Studies in judgment and decision making have found that people’s probability judgments can fail to be coherent. We investigated people’s coherence further for judgments about conjunctions, disjunctions and conditionals, and asked whether their coherence would increase when they were given the explicit task of drawing inferences. Participants gave confidence judgments about a list of separate statements (the statements group) or the statements grouped as explicit inferences (the inferences group). Their responses were generally coherent at above chance levels for all the inferences investigated, regardless of the presence of an explicit inference task. An exception was that they were incoherent in the context known to cause the conjunction fallacy, and remained so even when they were given an explicit inference. The participants were coherent under the assumption that they interpreted the natural language conditional as it is represented in Bayesian accounts of conditional reasoning, but they were incoherent under the assumption that they interpreted the natural language conditional as the material conditional of elementary binary logic. Our results provide further support for the descriptive adequacy of Bayesian reasoning principles in the study of deduction under uncertainty.

7,052 views
51 citations
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