My August 23, 2021 issue of The New Yorker (the cover of which is pictured above) arrived in the mail on Friday, and among the essays that caught my attention was this one on mental models and Bayesian reasoning by Joshua Rothman, who is the “ideas editor” for that fabled publication. The reason why I am pointing out Rothman’s excellent piece here is that it contains one of the best explanations of Bayesian reasoning that I have read since my own “conversion” to subjective probability in Amsterdam during the summer of 2011. Since that fateful moment, most of my scholarly work has been devoted to extending Bayesian reasoning into the domains of law and adjudication. Also, Bayesian reasoning is central to what I have been trying to do on this blog since I first began blogging in July of 2013. Below the fold, then, are a few excerpts — seven paragraphs in all — from the online version of Rothman’s beautiful essay (links in the original; emphasis added by me):
“A rational person must practice what the neuroscientist Stephen Fleming, in Know Thyself: The Science of Self-Awareness (Basic Books), calls ‘metacognition,’ or ‘the ability to think about our own thinking’—’a fragile, beautiful, and frankly bizarre feature of the human mind.’ ****
“Fleming notes that metacognition is a skill. Some people are better at it than others. [Julia Galef, the author of “The Scout Mindset: Why Some People See Things Clearly and Others Don’t” (Portfolio)] believes that, by ‘calibrating’ our metacognitive minds, we can improve our performance and so become more rational. In a section of her book called ‘Calibration Practice,’ she offers readers a collection of true-or-false statements (‘Mammals and dinosaurs coexisted’; ‘Scurvy is caused by a deficit of Vitamin C’); your job is to weigh in on the veracity of each statement while also indicating whether you are fifty-five, sixty-five, seventy-five, eighty-five, or ninety-five per cent confident in your determination. A perfectly calibrated individual, Galef suggests, will be right seventy-five per cent of the time about the answers in which she is seventy-five per cent confident. With practice, I got fairly close to ‘perfect calibration’: I still answered some questions wrong, but I was right about how wrong I would be.
“There are many calibration methods. In the ‘equivalent bet’ technique, which Galef attributes to the decision-making expert Douglas Hubbard, you imagine that you’ve been offered two ways of winning ten thousand dollars: you can either bet on the truth of some statement (for instance, that self-driving cars will be on the road within a year) or reach blindly into a box full of balls in the hope of retrieving a marked ball. Suppose the box contains four balls. Would you prefer to answer the question, or reach into the box? (I’d prefer the odds of the box.) Now suppose the box contains twenty-four balls—would your preference change? By imagining boxes with different numbers of balls, you can get a sense of how much you really believe in your assertions. For Galef, the box that’s ‘equivalent’ to her belief in the imminence of self-driving cars contains nine balls, suggesting that she has eleven-per-cent confidence in that prediction. Such techniques may reveal that our knowledge is more fine-grained than we realize; we just need to look at it more closely. Of course, we could be making out detail that isn’t there.
“There are many ways to explain Bayesian reasoning … but the basic idea is simple. When new information comes in, you don’t want it to replace old information wholesale. Instead, you want it to modify what you already know to an appropriate degree. The degree of modification depends both on your confidence in your preëxisting knowledge and on the value of the new data. Bayesian reasoners begin with what they call the ‘prior’ probability of something being true, and then find out if they need to adjust it.
“Consider the example of a patient who has tested positive for breast cancer—a textbook case used by Pinker and many other rationalists. The stipulated facts are simple. The prevalence of breast cancer in the population of women—the ‘base rate’—is one per cent. When breast cancer is present, the test detects it ninety per cent of the time. The test also has a false-positive rate of nine per cent: that is, nine per cent of the time it delivers a positive result when it shouldn’t. Now, suppose that a woman tests positive. What are the chances that she has cancer?
“When actual doctors answer this question, Pinker reports, many say that the woman has a ninety-per-cent chance of having it. In fact, she has about a nine-per-cent chance. The doctors have the answer wrong because they are putting too much weight on the new information (the test results) and not enough on what they knew before the results came in—the fact that breast cancer is a fairly infrequent occurrence. To see this intuitively, it helps to shuffle the order of your facts, so that the new information doesn’t have pride of place. Start by imagining that we’ve tested a group of a thousand women: ten will have breast cancer, and nine will receive positive test results. Of the nine hundred and ninety women who are cancer-free, eighty-nine will receive false positives. Now you can allow yourself to focus on the one woman who has tested positive. To calculate her chances of getting a true positive, we divide the number of positive tests that actually indicate cancer (nine) by the total number of positive tests (ninety-eight). That gives us about nine per cent. ****
“Bayesian reasoning implies a few ‘best practices.’ Start with the big picture, fixing it firmly in your mind. Be cautious as you integrate new information, and don’t jump to conclusions. Notice when new data points do and do not alter your baseline assumptions (most of the time, they won’t alter them), but keep track of how often those assumptions seem contradicted by what’s new. Beware the power of alarming news, and proceed by putting it in a broader, real-world context.”
Douglas Hofstadter explored some of this in his early work (Godel Escher Bach). The idea of “priors” and how they are considered mathematically compared to “presents” — well, I wonder whether the Bayes formula really respects indifference as to “pride of place”. I would have to delve into the math and logic of this, which, thanks to your blog post, I am inspired to do.
I need to go back and dust off my old copy of Hofstadter …