Overcoming difficulties in Bayesian reasoning: A reply to Lewis and Keren (1999) and Mellers and McGraw (1999).

Experts' and laypeople's reasoning in Bayesian-type problems can be improved by representing information in frequency formats rather than in probabilities. This thesis opens up important applications in medicine, law, statistics education, and other fields. The beneficial effect, which has been demonstrated in experimental and applied studies, seems to be no longer in dispute. Under dispute is what causes this effect and what its boundary conditions are. Lewis and Keren argue that the beneficial effect of frequency formats is due to "joint statements" rather than to "frequency statements". However, they overlook that our thesis is about frequency formats, not just any kind of frequency statements. We provide evidence that joint statements alone cannot account for the effect of frequency formats. Mellers and McGraw, in contrast, acknowledge the beneficial power of frequency formats but propose a boundary condition under which it is reduced. In a reanalysis of our original data, we find the specific interaction they report for the problem they used, but not for any other problem. After addressing these two commentaries, we summarize results indicating that teaching frequency representations fosters insight into Bayesian reasoning.