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To the Editor: In their letter, Cole et al1 set out to prove that a dogmatist (one whose beliefs cannot be influenced by observations) cannot learn (which they define as “having your beliefs influenced by observations”). Disregarding that this proposition is true by definition, the authors provided an amusing exploration into this idea using Bayes’ theorem. However, the authors made an important error in their calculations. This error does not change the authors’ point about dogmatists being unable to learn, but it does lead to the incorrect conclusion that it is good practice to allow any piece of information to sway your opinion, regardless of the quality of that information; this seems like a major oversight in this era of “fake news”2 and “alternative facts!”3,4 Cole et al1 consider an encrypted question that has two possible answers; we will denote these as or . The authors suppose that 12 people answer the question of interest and examine how prior belief in the correct answer is updated by these observations. The authors remind us of Bayes’ theorem, which says that belief in an answer after observing data [the posterior probability ] is a function of both prior belief and the likelihood of the observed data if the answer is . They define a dogmatic belief as one that is certain a priori [e.g., ] and observe that data do not influence this dogmatic belief . However, in their calculations, Cole et al1 conflated their answer to the question of interest with the probability that each person actually gave the right answer. In the remainder of this letter, I will correct the calculations and demonstrate the important finding that learning from bad information can be worse than not learning at all. As in reference (1), suppose that in absence of evidence, a nondogmatist believes the two possible answers to be equally likely . Cole et al1 wish to calculate the nondogmatist’s belief that after learning that five people think and seven think . That is, the goal is to find the posterior probability. The key point missed by Cole et al1 is that is a function of the probability (call it ) that each of these independent and exchangeable people gave the right answer. In their example with deterministic Q, If the observations came from 12 individuals who were each fairly likely to get the right answer, say = 2/3, then the posterior probabilities would be and , and as reported by Cole et al,1 a nondogmatist should be swayed to believe that the correct answer is . However, if the 12 “experts” were answering completely randomly (so = 1/2), then we can see that their information truly should not influence beliefs: and . Most importantly, if the information came from 12 people who were unlikely to give the right answer, say = 1/3, then beliefs should actually be influenced in the opposite direction: and . Without knowing the quality of the data, it is impossible to properly learn from it. If presented with bad data, the nondogmatist could easily be swayed to posterior beliefs that are further from the truth than the ones established a priori! In this light, I propose an addendum to the proposition by Cole et al1: it is true that dogmatists cannot learn, but learning may actually be detrimental to those who cannot separate truth from “alternative facts.” Michael A. McIsaac Department of Public Health Sciences Queen’s University Kingston, ON, Canada [email protected]

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