Why can it be so hard to solve Bayesian problems? Moving from number comprehension to relational reasoning demands

Abstract

Over the last decades, understanding the sources of the difficulty of Bayesian problem solving has been an important research goal, with the effects of numerical format and individual numeracy being widely studied. However, the focus on the comprehension of probability numbers has overshadowed the relational reasoning demand of the Bayesian task. This is particularly the case when the statistical data are verbally described since the requested quantitative relation (posterior ratio) is misaligned with the presented ones (prior and likelihood ratios). In this regard, here I develop the proposal that research on Bayesian reasoning might improve by considering the notational alignment framework of mathematical problem-solving. Specifically, this framework can help to understand the sources of the main difficulties underlying Bayesian inferences based on verbal descriptions. In essence, the present proposal supports the general claim in math education regarding the need to foster relational comprehension to avoid misleading alignments and improve problem solving.

Keywords:

• Bayesian reasoning
• mathematical word-problem solving
• relational reasoning
• structural mapping

Acknowledgments

I thank Aidan Feeney, Miroslav Sirota and one anonymous reviewer for the insightful and helpful comments on earlier drafts of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The present proposal does not focus on Bayesian problems with percentages or decimals because, in addition to the relational reasoning demand, difficulties in these problems are also due to the required computation. By contrast, problems that present and request natural frequencies only require adding two numbers. Understanding why these, computationally simple, problems remain difficult for many people is still an open issue.

2 Accuracy of the problems reported in Weber et al. (2018) were 51% and 22%. The hit rate of the problem with the lowest accuracy was “55 of 100” which, as found in other studies (see the main text), seemed to increase the tendency to use “100” as the denominator of the posterior ratio. By contrast, the hit rate of the problem with higher accuracy was “10 of 10”, which might have facilitated the selection of the numerator, reducing the mapping of 100 onto the denominator.

3 In Johnson and Tubau (2013), participants with a low level of numeracy were less accurate for problems including long texts than for problems with shorter text. In the latter case, the effect of numeracy was not significant.

4 Numeracy questionnaires such as the ones of Cokely et al. (2012), Peters et al. (2007) or Primi et al. (2017) also include Bayesian problems and, hence, based on the present proposal, they also may assess complex relational reasoning skills. However, here I focus on the assessment of numeracy through non-Bayesian statistical problems.

5 Results concerning numeracy were not reported in these studies. Nevertheless, the finding that performance was as low as for normalized data suggests that natural frequency problems requesting a single-event posterior probability are similarly difficult, regardless the level of numeracy.

6 For example, Martin and Bassok (2005) observed that the inverse error in algebraic modeling (see section 2.1) was less frequent for statements referring to asymmetrical relations (e.g., “At a certain university, there are 6 times as many students as professors”) than for those referring to symmetrical relations (e.g., “A certain factory produces 6 times as many nails as screws”). In the asymmetrical context, it was easier to use division and the corresponding correct algebraic model.