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

Abstract

Presenting natural frequencies facilitates Bayesian inferences compared to using percentages.Nevertheless, many people, including highly educated and skilled reasoners, still fail to provide Bayesian responses on these computationally simple problems. We show that the complexity of relational reasoning (e.g. structural mapping between presented and requested relations), can help explain remaining difficulties. With a non-Bayesian inference which required identical arithmetic but afforded more direct structural mapping, performance was universally high.Furthermore, reducing the relational demands of the task with questions which directed reasoners to use the presented statistics, compared with questions which prompted the representation of a second, similar sample, significantly improved reasoning. Distinct error patterns were also observed between these presented-and similar-sample scenarios, which suggested differences in relational reasoning strategies. On the other hand, while higher numeracy was associated with better Bayesian reasoning, higher numerate reasoners were not immune to the relational complexity of the task. Together, these findings validate the relational reasoning view of Bayesian problem solving, and highlight the importance of considering not only the presented task structure, but also the complexity of the structural alignment between presented and requested relations.

From reading numbers to seeing ratios: a benefit of icons for risk comprehension

“…These findings support the hypothesis that icons facilitate the comprehension of the ratio beyond the represented quantities. Furthermore, the large distribution of errors in both verbal formats confirmed the suggestion that verbal presentations induce superficial reasoning and misleading associations (Barbey & Sloman, 2007;Johnson & Tubau, 2017). Results also suggest that natural frequencies, like percentages, are unhelpful for inferring single-event probabilities.…”

“…Results also suggest that natural frequencies, like percentages, are unhelpful for inferring single-event probabilities. Nevertheless, as previously shown in the context of frequency estimates (Johnson & Tubau, 2017), this limitation might be related to the misalignment of the Bayesian inference. Experiment 2 aimed to test this hypothesis.…”

“…As suggested recently by Johnson & Tubau (2017), Bayesian word-problems reporting natural frequencies can be as difficult as the ones reporting percentages because they do not eliminate the relational misalignment between presented and requested relations. More specifically, in standard verbal presentations, numbers are associated with either the set or the subset of the relationship.…”

“…Accordingly, number-relational role associations induced from standard presentations make it possible to use such numbers in a similar role ("of 4+96 women, 3+12 receive a positive mammogram"), but hinder their use in a different role, as required in the Bayesian inference "of 3+12 women with positive mammogram, 3 have breast cancer," where the bold part indicates the set (posterior reference class in Figure 1). In this regard, inaccurate Bayesian reasoning might be caused not only by a limited understanding of the nested-set structure of the data (Barbey & Sloman, 2007), or how to translate frequencies into probabilities (Cosmides & Tooby, 1996;Gigerenzer, 1991), but by difficulties involved in mapping the presented relationships onto the requested one (preliminary evidences supporting this proposal, using natural frequencies, are reported in Johnson & Tubau, 2017). Figure 1 Interestingly, and also consistent with this proposal, problems that present the sample statistics as frequency grids or icon arrays make it easier to solve the Bayesian question in frequency format, compared to verbal presentations alone (e.g., Brase, 2009 and2014;Galesic, Garcia-Retamero & Gigerenzer, 2009;Garcia-Retamero & Hoffrage, 2013;Sedlmeier & Gigerenzer, 2001; but see Brase & Hill, 2017;Cosmides &Tooby, 1996 andSirota et al, 2014b for mixed findings).…”

Numeracy and Risk Literacy: What Have We Learned so Far?

“…Patients with low numeracy also make less accurate diagnostic inferences based on numerical information about screening (García-Retamero, Cokely, & Hoffrage, 2015; García-Retamero & Hoffrage, 2013; Gigerenzer, Gaissmaier, Kurz-Milcke, Schwartz, & Woloshin, 2007; Petrova, García-Retamero, Catena, & van der Pligt, 2016) and are less able to use this diagnostic information to adjust their risk estimates (Schwartz et al, 1997). Research on the underlying mechanisms shows that people with low numeracy have difficulties reasoning about the underlying relationships in the data (Johnson & Tubau, 2017) and do no benefit from interventions that clarify the causal structure of the data (McNair & Feeney, 2015). Finally, less numerate patients are more easily biased by the way health-related numerical information is framed (García-Retamero & Galesic, 2009, 2010, 2011; Peters & Levin, 2008; Peters et al, 2006).…”