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# December 2019

Personal Blogposts
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53Buck2mo[I'm not sure how good this is, it was interesting to me to think about, idk if it's useful, I wrote it quickly.] Over the last year, I internalized Bayes' Theorem much more than I previously had; this led me to noticing that when I applied it in my life it tended to have counterintuitive results; after thinking about it for a while, I concluded that my intuitions were right and I was using Bayes wrong. (I'm going to call Bayes' Theorem "Bayes" from now on.) Before I can tell you about that, I need to make sure you're thinking about Bayes in terms of ratios rather than fractions. Bayes is enormously easier to understand and use when described in terms of ratios. For example: Suppose that 1% of women have a particular type of breast cancer, and a mammogram is 20 times more likely to return a positive result if you do have breast cancer, and you want to know the probability that you have breast cancer if you got that positive result. The prior probability ratio is 1:99, and the likelihood ratio is 20:1, so the posterior probability is 1∗20:99∗1 = 20:99, so you have probability of 20/(20+99) of having breast cancer. I think that this is absurdly easier than using the fraction formulation. I think that teaching the fraction formulation is the single biggest didactic mistake that I am aware of in any field. -------------------------------------------------------------------------------- Anyway, a year or so ago I got into the habit of calculating things using Bayes whenever they came up in my life, and I quickly noticed that Bayes seemed surprisingly aggressive to me. For example, the first time I went to the Hot Tubs of Berkeley, a hot tub rental place near my house, I saw a friend of mine there. I wondered how regularly he went there. Consider the hypotheses of "he goes here three times a week" and "he goes here once a month". The likelihood ratio is about 12x in favor of the former hypothesis. So if I previously was ten to one against the three-times-a-week hyp