Effects of visualizing statistical information - an empirical study on tree diagrams and 2 × 2 tables

The impact of problem domain on Bayesian inferences: A systematic investigation

Sparse (and occasionally contradictory) evidence exists regarding the impact of domain on probabilistic updating, some of which suggests that Bayesian word problems with medical content may be especially challenging. The present research aims to address this gap in knowledge through three pre-registered online studies, which involved a total of 2,238 participants. Bayesian word problems were related to one of three domains: medical, daily-life, and abstract. In the first two cases, problems presented realistic content and plausible numerical information, while in the latter, problems contained explicitly imaginary elements. Problems across domains were matched in terms of all relevant statistical values and, as much as possible, wording. Studies 1 and 2 utilized the same set of problems, but different response elicitation methods (i.e., an open-ended and a multiple-choice question, respectively). Study 3 involved a larger number of participants per condition and a smaller set of problems to more thoroughly investigate the magnitude of differences between the domains. There was a generally low rate of correct responses (17.2%, 17.4%, and 14.3% in Studies 1, 2, and 3, respectively), consistent with accuracy levels commonly observed in the literature for this specific task with online samples. Nonetheless, a small but significant difference between domains was observed: participants' accuracy did not differ between medical and daily-life problems, while it was significantly higher in corresponding abstract problems. These results suggest that medical problems are not inherently more difficult to solve, but rather that performance is improved with abstract problems for which participants cannot draw from their background knowledge.

How general is the natural frequency effect? The case of joint probabilities

Natural frequencies are known to improve performance in Bayesian reasoning. However, their impact in situations with two binary events has not yet been completely examined, as most researchers in the last 30 years focused only on conditional probabilities. Nevertheless, situations with two binary events consist of 16 elementary probabilities and so we widen the scope and focus on joint probabilities. In this article, we theoretically elaborate on the importance of joint probabilities, for example, in situations like the Linda problem. Furthermore, we implemented a study in a 2×5×2 design with the factors information format (probabilities vs. natural frequencies), visualization type ("Bayesian text" vs. tree diagram vs. double tree diagram vs. net diagram vs. 2×2 table), and context (mammography vs. economics problem). Additionally, all four "joint questions" (i.e., P ( A ∩ B ) , P ( A ¯ ∩ B ) , P ( A ¯ ∩ B ¯ ) , P ( A ∩ B ¯ ) ) were asked for. The main factor of interest was whether there is a format effect in the five visualization types named above. Surprisingly, the advantage of natural frequencies was not found for joint probabilities and, most strikingly, the format interacted with the visualization type. Specifically, while people's understanding of joint probabilities in a double tree seems to be worse than the understanding of the corresponding natural frequencies (and, thus, the frequency effect holds true), the opposite seems to be true in the 2 × 2 table. Hence, the advantage of natural frequencies compared to probabilities in typical Bayesian tasks cannot be found in the same way when joint probability or frequency tasks are asked.

Measuring people's covariational reasoning in Bayesian situations

Previous research on Bayesian reasoning has typically investigated people's ability to assess a posterior probability (i.e., a positive predictive value) based on prior knowledge (i.e., base rate, true-positive rate, and false-positive rate). In this article, we systematically examine the extent to which people understand the effects of changes in the three input probabilities on the positive predictive value, that is, covariational reasoning. In this regard, two different operationalizations for measuring covariational reasoning (i.e., by single-choice vs. slider format) are investigated in an empirical study with N = 229 university students. In addition, we aim to answer the question wheter a skill in "conventional" Bayesian reasoning is a prerequisite for covariational reasoning.

Analyzing and supporting mental representations and strategies in solving Bayesian problems

Solving Bayesian problems poses many challenges, such as identifying relevant numerical information, classifying, and translating it into mathematical formula language, and forming a mental representation. This triggers research on how to support the solving of Bayesian problems. The facilitating effect of using numerical data in frequency format instead of probabilities is well documented, as is the facilitating effect of given visualizations of statistical data. The present study not only compares the visualizations of the 2 × 2 table and the unit square, but also focuses on the results obtained from the self-creation of these visualizations by the participants. Since it has not yet been investigated whether the better correspondence between external and internal visualization also has an effect on cognitive load when solving Bayesian tasks, passive and active cognitive load are additionally measured. Due to the analog character and the proportional representation of the numerical information by the unit square, it is assumed that the passive cognitive load is lower when using the unit square as visualization than when using the 2 × 2 table. The opposite is true for active cognitive load.

Heterogeneity of rules in Bayesian reasoning: A toolbox analysis

How do people infer the Bayesian posterior probability from stated base rate, hit rate, and false alarm rate? This question is not only of theoretical relevance but also of practical relevance in medical and legal settings. We test two competing theoretical views: single-process theories versus toolbox theories. Single-process theories assume that a single process explains people's inferences and have indeed been observed to fit people's inferences well. Examples are Bayes's rule, the representativeness heuristic, and a weighing-and-adding model. Their assumed process homogeneity implies unimodal response distributions. Toolbox theories, in contrast, assume process heterogeneity, implying multimodal response distributions. After analyzing response distributions in studies with laypeople and professionals, we find little support for the single-process theories tested. Using simulations, we find that a single process, the weighing-and-adding model, nevertheless can best fit the aggregate data and, surprisingly, also achieve the best out-of-sample prediction even though it fails to predict any single respondent's inferences. To identify the potential toolbox of rules, we test how well candidate rules predict a set of over 10,000 inferences (culled from the literature) from 4,188 participants and 106 different Bayesian tasks. A toolbox of five non-Bayesian rules plus Bayes's rule captures 64% of inferences. Finally, we validate the Five-Plus toolbox in three experiments that measure response times, self-reports, and strategy use. The most important conclusion from these analyses is that the fitting of single-process theories to aggregate data risks misidentifying the cognitive process. Antidotes to that risk are careful analyses of process and rule heterogeneity across people.

Boosting for insight and/or boosting for agency? How to maximize accurate test interpretation with natural frequencies

BACKGROUND

Many physicians do not know how to accurately interpret test results using Bayes' rule. As a remedy, two kinds of interventions have been shown effective: boosting insight and boosting agency with natural frequencies. To boost insight, test statistics are provided in natural frequencies (rather than conditional probabilities), without instructions on how to use them. To boost agency, a training is provided on how to translate probabilities into natural frequencies and apply them in Bayes' rule. What has not been shown is whether boosting agency is sufficient or if representing test statistics in natural frequencies may additionally boost insight to maximize accurate test interpretation.

METHODS

We used a pre/posttest design to assess test interpretation accuracy of 577 medical students before and after a training on two Bayesian reasoning tasks, one providing conditional probabilities, the other natural frequencies. The pretest assessed baseline abilities versus the effect of natural frequencies to boost insight. After participants received a training on how to translate conditional probabilities into natural frequencies and how to apply them in Bayes' rule, test interpretation skills were assessed using the same tasks again, comparing the effects of training-induced agency with versus without additionally boosting insight (i.e., test statistics in natural frequencies versus conditional probabilities).

RESULTS

Compared to the test question formatted in conditional probabilities (34% correct answers), natural frequencies facilitated Bayesian reasoning without training (68%), that is, they increased insight. The training on how to use natural frequencies improved performance for tasks formatted in conditional probabilities (64%). Performance was maximal after training and with test statistics formatted in natural frequencies, that is, with a combination of boosting insight and agency (89%).

CONCLUSIONS

Natural frequencies should be used to boost insight and agency to maximize effective use of teaching resources. Thus, mandating that test statistics are provided in natural frequencies and adopting short trainings on how to translate conditional probabilities into natural frequencies and how to apply them in Bayes' rule will help to maximize accurate test interpretation.

TRIAL REGISTRATION

The study was a registered with the German Clinical Trial Registry ( DRKS00008723 ; 06/03/2015).

How to Train Novices in Bayesian Reasoning

Bayesian Reasoning is both a fundamental idea of probability and a key model in applied sciences for evaluating situations of uncertainty. Bayesian Reasoning may be defined as the dealing with, and understanding of, Bayesian situations. This includes various aspects such as calculating a conditional probability (performance), assessing the effects of changes to the parameters of a formula on the result (covariation) and adequately interpreting and explaining the results of a formula (communication). Bayesian Reasoning is crucial in several non-mathematical disciplines such as medicine and law. However, even experts from these domains struggle to reason in a Bayesian manner. Therefore, it is desirable to develop a training course for this specific audience regarding the different aspects of Bayesian Reasoning. In this paper, we present an evidence-based development of such training courses by considering relevant prior research on successful strategies for Bayesian Reasoning (e.g., natural frequencies and adequate visualizations) and on the 4C/ID model as a promising instructional approach. The results of a formative evaluation are described, which show that students from the target audience (i.e., medicine or law) increased their Bayesian Reasoning skills and found taking part in the training courses to be relevant and fruitful for their professional expertise.

Natural frequency trees improve diagnostic efficiency in Bayesian reasoning

When physicians are asked to determine the positive predictive value from the a priori probability of a disease and the sensitivity and false positive rate of a medical test (Bayesian reasoning), it often comes to misjudgments with serious consequences. In daily clinical practice, however, it is not only important that doctors receive a tool with which they can correctly judge-the speed of these judgments is also a crucial factor. In this study, we analyzed accuracy and efficiency in medical Bayesian inferences. In an empirical study we varied information format (probabilities vs. natural frequencies) and visualization (text only vs. tree only) for four contexts. 111 medical students participated in this study by working on four Bayesian tasks with common medical problems. The correctness of their answers was coded and the time spent on task was recorded. The median time for a correct Bayesian inference is fastest in the version with a frequency tree (2:55 min) compared to the version with a probability tree (5:47 min) or to the text only versions based on natural frequencies (4:13 min) or probabilities (9:59 min).The score diagnostic efficiency (calculated by: median time divided by percentage of correct inferences) is best in the version with a frequency tree (4:53 min). Frequency trees allow more accurate and faster judgments. Improving correctness and efficiency in Bayesian tasks might help to decrease overdiagnosis in daily clinical practice, which on the one hand cause cost and on the other hand might endanger patients' safety.

Tversky and Kahneman's Cognitive Illusions: Who Can Solve Them, and Why?

In the present paper we empirically investigate the psychometric properties of some of the most famous statistical and logical cognitive illusions from the "heuristics and biases" research program by Daniel Kahneman and Amos Tversky, who nearly 50 years ago introduced fascinating brain teasers such as the famous Linda problem, the Wason card selection task, and so-called Bayesian reasoning problems (e.g., the mammography task). In the meantime, a great number of articles has been published that empirically examine single cognitive illusions, theoretically explaining people's faulty thinking, or proposing and experimentally implementing measures to foster insight and to make these problems accessible to the human mind. Yet these problems have thus far usually been empirically analyzed on an individual-item level only (e.g., by experimentally comparing participants' performance on various versions of one of these problems). In this paper, by contrast, we examine these illusions as a group and look at the ability to solve them as a psychological construct. Based on an sample of N = 2,643 Luxembourgian school students of age 16-18 we investigate the internal psychometric structure of these illusions (i.e., Are they substantially correlated? Do they form a reflexive or a formative construct?), their connection to related constructs (e.g., Are they distinguishable from intelligence or mathematical competence in a confirmatory factor analysis?), and the question of which of a person's abilities can predict the correct solution of these brain teasers (by means of a regression analysis).

Perspectives on the 2 × 2 Matrix: Solving Semantically Distinct Problems Based on a Shared Structure of Binary Contingencies

Cognition is both empowered and limited by representations. The matrix lens model explicates tasks that are based on frequency counts, conditional probabilities, and binary contingencies in a general fashion. Based on a structural analysis of such tasks, the model links several problems and semantic domains and provides a new perspective on representational accounts of cognition that recognizes representational isomorphs as opportunities, rather than as problems. The shared structural construct of a 2 × 2 matrix supports a set of generic tasks and semantic mappings that provide a unifying framework for understanding problems and defining scientific measures. Our model's key explanatory mechanism is the adoption of particular perspectives on a 2 × 2 matrix that categorizes the frequency counts of cases by some condition, treatment, risk, or outcome factor. By the selective steps of filtering, framing, and focusing on specific aspects, the measures used in various semantic domains negotiate distinct trade-offs between abstraction and specialization. As a consequence, the transparent communication of such measures must explicate the perspectives encapsulated in their derivation. To demonstrate the explanatory scope of our model, we use it to clarify theoretical debates on biases and facilitation effects in Bayesian reasoning and to integrate the scientific measures from various semantic domains within a unifying framework. A better understanding of problem structures, representational transparency, and the role of perspectives in the scientific process yields both theoretical insights and practical applications.

Different Visualizations Cause Different Strategies When Dealing With Bayesian Situations

People often struggle with Bayesian reasoning. However, previous research showed that people's performance (and rationality) can be supported by the way the statistical information is represented. First, research showed that using natural frequencies instead of probabilities as the format of statistical information significantly increases people's performance in Bayesian situations. Second, research also revealed that people's performance increases through using visualization. We have built our paper on existing research in this field. Our main aim was to analyze people's strategies in Bayesian situations that are erroneous even though statistical information is represented as natural frequencies and visualizations. In particular, we compared two pairs of visualization with similar numerical information (tree diagram vs. unit square, and double-tree diagram vs. 2 × 2-table) concerning their impact on people's erroneous strategies in Bayesian situations. For this aim, we conducted an experiment with 540 university students. The students were randomly assigned to four conditions defined by the four different visualizations of statistical information. The students were asked to indicate a fraction in response to four Bayesian situations. We documented the numerator and denominator of the students' responses representing a basic set and a subset in a Bayesian situation. Our results showed that people's erroneous strategies are highly dependent on visualization. A central finding was that the visualization's characteristic of making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people's Bayesian reasoning. For example, compared to the unit square, a tree diagram does not explicitly visualize the set-subset relations that are relevant in a Bayesian situation. Accordingly, compared to a unit square, a tree diagram partly hinders people in finding the correct denominator in a Bayesian situation, and, in particular, triggers selecting a wrong numerator. By analyzing people's erroneous strategies in Bayesian situations, we contribute to investigating approaches to facilitate Bayesian reasoning and to further develop the teaching of Bayesian reasoning.

The Effect of Information Formats and Incidental Affect on Prior and Posterior Probability Judgments

Background. Interpreting medical test results involves judging probabilities, including making Bayesian inferences such as judging the positive and negative predictive values. Although prior work has shown that experience formats (e.g., slide shows of representative patient cases) produce more accurate Bayesian inferences than description formats (e.g., verbal statistical summaries), there are disadvantages of using the experience format for real-world medical decision making that may be solved by presenting relevant information in a 2 × 2 table format. Furthermore, medical decisions are often made in stressful contexts, yet little is known about the influence of acute stress on the accuracy of Bayesian inferences. This study aimed to a) replicate the description-experience format effect on probabilistic judgments; b) examine judgment accuracy across description, experience, and a new 2 × 2 table format; and c) assess the effect of acute stress on probability judgments. Method. The study employed a 2 (stress condition) × 3 (format) factorial between-subjects design. Participants (N = 165) completed a Bayesian inference task in which information about a medical screening test was presented in 1 of 3 formats (description, experience, 2 × 2 table), following a laboratory stress induction or a no-stress control condition. Results. Overall, the 2 × 2 table format produced the most accurate probability judgments, including Bayesian inferences, compared with the description and experience formats. Stress had no effect on judgment accuracy. Discussion. Given its accuracy and practicality, a 2 × 2 table may be better suited than description or experience formats for communicating probabilistic information in medical contexts.

A New Visualization for Probabilistic Situations Containing Two Binary Events: The Frequency Net

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2 × 2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch structures, these two visualizations (in contrast to the 2 × 2 table) can even simultaneously display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2 × 2 tables and (double-)trees have a decisive disadvantage: While joint probabilities [e.g., P(A∩B)] are represented in 2 × 2 tables but no conditional probabilities [e.g., P(A|B)], it is exactly the other way around with (double-)trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2 × 2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2 × 2 table, double-tree, or net diagram with absolute frequencies) are consistently superior to probability visualizations, and the frequency net performs as well as the frequency double-tree. Only the 2 × 2 table with frequencies—the one visualization that participants were already familiar with—led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, but no uniform format effect becomes visible. Here, participants reached the highest performance in the versions with probability 2 × 2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability.

Evidencing How Experience and Problem Format Affect Probabilistic Reasoning Through Interaction Analysis

This paper examines the role that lived experience plays in the human capacity to reason about uncertainty. Previous research shows that people are more likely to provide accurate responses in Bayesian tasks when the data are presented in natural frequencies, the problem in question describes a familiar event, and the values of the data are in line with beliefs. Precisely why these factors are important remains open to debate. We elucidate the issue in two ways. Firstly, we hypothesize that in a task that requires people to reason about conditional probabilities, they are more likely to respond accurately when the values of the problem reflect their own lived experience, than when they reflect the experience of the average participant. Secondly, to gain further understanding of the underlying reasoning process, we employ a novel interaction analysis method that tracks mouse movements in an interactive web application and applies transition analysis to model how the approach to reasoning differs depending on whether data are presented using percentages or natural frequencies. We find (1) that the closer the values of the data in the problem are to people's self-reported lived experience, the more likely they are to provide a correct answer, and (2) that the reasoning process employed when data are presented using natural frequencies is qualitatively different to that employed when data are presented using percentages. The results indicate that the benefits of natural frequency presentation are due to a clearer representation of the relationship between sets and that the prior humans acquire through experience has an overwhelming influence on their ability to reason about uncertainty.

An Eye-Tracking Study of Statistical Reasoning With Tree Diagrams and 2 × 2 Tables

Changing the information format from probabilities into frequencies as well as employing appropriate visualizations such as tree diagrams or 2 × 2 tables are important tools that can facilitate people's statistical reasoning. Previous studies have shown that despite their widespread use in statistical textbooks, both of those visualization types are only of restricted help when they are provided with probabilities, but that they can foster insight when presented with frequencies instead. In the present study, we attempt to replicate this effect and also examine, by the method of eye tracking, why probabilistic 2 × 2 tables and tree diagrams do not facilitate reasoning with regard to Bayesian inferences (i.e., determining what errors occur and whether they can be explained by scan paths), and why the same visualizations are of great help to an individual when they are combined with frequencies. All ten inferences of N = 24 participants were based solely on tree diagrams or 2 × 2 tables that presented either the famous "mammography context" or an "economics context" (without additional textual wording). We first asked participants for marginal, conjoint, and (non-inverted) conditional probabilities (or frequencies), followed by related Bayesian tasks. While solution rates were higher for natural frequency questions as compared to probability versions, eye-tracking analyses indeed yielded noticeable differences regarding eye movements between correct and incorrect solutions. For instance, heat maps (aggregated scan paths) of distinct results differed remarkably, thereby making correct and faulty strategies visible in the line of theoretical classifications. Moreover, the inherent structure of 2 × 2 tables seems to help participants avoid certain Bayesian mistakes (e.g., "Fisherian" error) while tree diagrams seem to help steer them away from others (e.g., "joint occurrence"). We will discuss resulting educational consequences at the end of the paper.

How to Improve Performance in Bayesian Inference Tasks: A Comparison of Five Visualizations

Bayes’ formula is a fundamental statistical method for inference judgments in uncertain situations used by both laymen and professionals. However, since people often fail in situations where Bayes’ formula can be applied, how to improve their performance in Bayesian situations is a crucial question. We based our research on a widely accepted beneficial strategy in Bayesian situations, representing the statistical information in the form of natural frequencies. In addition to this numerical format, we used five visualizations: a 2 × 2-table, a unit square, an icon array, a tree diagram, and a double-tree diagram. In an experiment with 688 undergraduate students, we empirically investigated the effectiveness of three graphical properties of visualizations: area-proportionality, use of discrete and countable statistical entities, and graphical transparency of the nested-sets structure. We found no additional beneficial effect of area proportionality. In contrast, the representation of discrete objects seems to be beneficial. Furthermore, our results show a strong facilitating effect of making the nested-sets structure of a Bayesian situation graphically transparent. Our results contribute to answering the questions of how and why a visualization could facilitate judgment and decision making in situations of uncertainty.

Why Can Only 24% Solve Bayesian Reasoning Problems in Natural Frequencies: Frequency Phobia in Spite of Probability Blindness

For more than 20 years, research has proven the beneficial effect of natural frequencies when it comes to solving Bayesian reasoning tasks (Gigerenzer and Hoffrage, 1995). In a recent meta-analysis, McDowell and Jacobs (2017) showed that presenting a task in natural frequency format increases performance rates to 24% compared to only 4% when the same task is presented in probability format. Nevertheless, on average three quarters of participants in their meta-analysis failed to obtain the correct solution for such a task in frequency format. In this paper, we present an empirical study on what participants typically do wrong when confronted with natural frequencies. We found that many of them did not actually use natural frequencies for their calculations, but translated them back into complicated probabilities instead. This switch from the intuitive presentation format to a less intuitive calculation format will be discussed within the framework of psychological theories (e.g., the Einstellung effect).

Visualizing the Bayesian 2-test case: The effect of tree diagrams on medical decision making

In medicine, diagnoses based on medical test results are probabilistic by nature. Unfortunately, cognitive illusions regarding the statistical meaning of test results are well documented among patients, medical students, and even physicians. There are two effective strategies that can foster insight into what is known as Bayesian reasoning situations: (1) translating the statistical information on the prevalence of a disease and the sensitivity and the false-alarm rate of a specific test for that disease from probabilities into natural frequencies, and (2) illustrating the statistical information with tree diagrams, for instance, or with other pictorial representation. So far, such strategies have only been empirically tested in combination for “1-test cases”, where one binary hypothesis (“disease” vs. “no disease”) has to be diagnosed based on one binary test result (“positive” vs. “negative”). However, in reality, often more than one medical test is conducted to derive a diagnosis. In two studies, we examined a total of 388 medical students from the University of Regensburg (Germany) with medical “2-test scenarios”. Each student had to work on two problems: diagnosing breast cancer with mammography and sonography test results, and diagnosing HIV infection with the ELISA and Western Blot tests. In Study 1 (N = 190 participants), we systematically varied the presentation of statistical information (“only textual information” vs. “only tree diagram” vs. “text and tree diagram in combination”), whereas in Study 2 (N = 198 participants), we varied the kinds of tree diagrams (“complete tree” vs. “highlighted tree” vs. “pruned tree”). All versions were implemented in probability format (including probability trees) and in natural frequency format (including frequency trees). We found that natural frequency trees, especially when the question-related branches were highlighted, improved performance, but that none of the corresponding probabilistic visualizations did.

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

It is an ongoing debate, what properties of visualizations increase people's performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes' rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it only in a numerical way. Accordingly, in a first experiment with 148 university students, the unit square outperformed the tree diagram when referring to the students' ability to quantify the subset relation that must be applied in Bayes' rule. As hypothesized, in a second experiment with 143 students, the unit square was significantly more effective when the students' performance in tasks based on Bayes' rule was regarded. Our results could inform the debate referring to Bayesian reasoning since we found that the graphical transparency of nested sets could explain these visualizations' effect.

Natural frequencies improve Bayesian reasoning in simple and complex inference tasks

Representing statistical information in terms of natural frequencies rather than probabilities improves performance in Bayesian inference tasks. This beneficial effect of natural frequencies has been demonstrated in a variety of applied domains such as medicine, law, and education. Yet all the research and applications so far have been limited to situations where one dichotomous cue is used to infer which of two hypotheses is true. Real-life applications, however, often involve situations where cues (e.g., medical tests) have more than one value, where more than two hypotheses (e.g., diseases) are considered, or where more than one cue is available. In Study 1, we show that natural frequencies, compared to information stated in terms of probabilities, consistently increase the proportion of Bayesian inferences made by medical students in four conditions-three cue values, three hypotheses, two cues, or three cues-by an average of 37 percentage points. In Study 2, we show that teaching natural frequencies for simple tasks with one dichotomous cue and two hypotheses leads to a transfer of learning to complex tasks with three cue values and two cues, with a proportion of 40 and 81% correct inferences, respectively. Thus, natural frequencies facilitate Bayesian reasoning in a much broader class of situations than previously thought.