Haynes Goddard writes:
I thought to do some reading in psychology on why Bayesian probability seems so counterintuitive, and making it difficult for many to learn and apply. Indeed, that is the finding of considerable research in psychology. It turns out that it is counterintuitive because of the way it is presented, following no doubt the way the textbooks are written. The theorem is usually expressed first with probabilities instead of frequencies, or “natural numbers” – counts in the binomial case.
The literature is considerable, starting at least with a seminal piece by David Eddy (1982). “Probabilistic reasoning in clinical medicine: problems and opportunities,” in Judgment under Uncertainty: Heuristics and Biases, eds D. Kahneman, P. Slovic and A. Tversky. Also much cited are Gigerenzer and Hoffrage (1995) “How to improve Bayesian reasoning without instruction: frequency formats” Psychol. Rev, and also Cosmides and Tooby, “Are humans good intuitive statisticians after all? Rethinking some conclusions from the literature on judgment under uncertainty”, Cognition, 1996.
This literature has amply demonstrated that people actually can readily and accurately reason in Bayesian terms if the data are presented in frequency form, but have difficulty if the data are given as percentages or probabilities. Cosmides and Tooby argue that this is so for evolutionary reasons, and their argument seems compelling.
So taking a look at my several texts (not a random sample of course), including Andrew’s well written text, I wanted to know how many authors introduce the widely used Bayesian example of determining the posterior probability of breast cancer after a positive mammography in numerical frequency terms or counts first, then shifting to probabilities. None do, although some do provide an example in frequency terms later.
Assuming that my little convenience sample is somewhat representative, it raises the question of why are not the recommendations of the psychologists adopted.
This is a missed opportunity, as the psychological findings indicate that the frequency approach makes Bayesian logic instantly clear, making it easier to comprehend the theorem in probability terms.
Since those little medical inference problems are very compelling, it would make the lives of a lot of students a lot easier and increase acceptance of the approach. One can only imagine how much sooner the sometimes acrimonious debates between frequentists and Bayesians would have diminished if not ended. So there is a clear lesson here for instructors and textbook writers.
Here is an uncommonly clear presentation of the breast cancer example: http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/. And there are numerous comments from beginning statistics students noting this clarity.
I agree, and in a recent introductory course I prepared, I did what you recommend and started right away with frequencies, Gigerenzer-style.
Why has it taken us so long to do this? I dunno, force of habit, I guess? I am actually pretty proud of chapter 1 of BDA (especially in the 3rd edition with its new spell-checking example, but even all the way back to the 1st edition in 1995) in that we treat probability as a quantity that can be measured empirically, and we avoid what I see as the flaw of seeking a single foundational justification for probability. Probability is a mathematical model with many different applications, including frequencies, prediction, betting, etc. There’s no reason to think of any one of these applications as uniquely fundamental.
But, yeah, I agree it would be better to start with the frequency calculations: instead of “1% probability,” talk about 10 cases out of 1000, etc.
P.S. It’s funny that Goddard cited a paper by Cosmides and Tooby, as they’re coauthors on that notorious fat-arms-and-political-attitudes paper, a recent gem in the garden-of-forking-paths, power=.06 genre. Nobody’s perfect, I guess. In particular, it’s certainly possible for people to do good research on the teaching and understanding of statistics, even while being confused about some key statistical principles themselves. And even the legendary Kahneman has been known, on occasion, to overstate the strength of statistical evidence.