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The cult of Bayes

April 7, 2014

Every so often I wander into a dark corner of the internet and find myself surrounded by creatures who speak to one another at length in a language I can hardly recognize. Except that’s not right–libertarians, skeptics, and transhumanists all speak perfectly fine English; I just can’t quite understand what would lead them to say the things they say, or to care about the things they care about. At least not with some effort. Though there was a time when I, too, spoke a dialect of nerd, and I can still sympathize with those who speak the lingo of open source, speculative fiction, and mathematics.

I found myself in such a dark wood this weekend; specifically, it was the dark wood of LessWrong, “a community blog devoted to refining the art of human rationality.” It’s essentially a bunch of traditional rationalists who have come to the realization that human minds are really terrible at being rational, and who are trying to develop strategies 1) to cope with that fact and 2) to bring on the Singularity in the form of Friendly AI. Their cult leader is a fellow named Eliezer Yudkowsky, who has a number of interestingly, ultimately wrong thoughts about how to define intelligence. Strangely, he also writes a Harry Potter fanfiction in which Harry was raised to worship the scientific method. Surprisingly, it’s worth reading–in fact, the world-building is better than in J.K. Rowling’s version. I suspect Eliezer intended to write Harry as a Mary Sue, but somewhat missed his target due to his philosophy being in fact rather tragically pathetic to anyone with a decent sense of the postlapsarian condition.

G.K. Chesterton wrote that “Every heresy is a truth taught out of proportion.” Eliezer Yudkowsky overstates the importance of Bayes’ Theorem, but he’s right that it’s quite important, and he’s right that it’s difficult to make intuitive. I encourage everyone, even (especially?) those not mathematically inclined, to read his introduction to Bayesian probability theory. Otherwise, you will answer the following question wrongly. You won’t be alone–“studies show” 85% of people answer wrongly–but you’ll still be wrong.

Here’s a story problem about a situation that doctors often encounter:

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

Most people say a number between seventy and eighty percent, that is, three out of every four, but the correct answer is more like one out of every thirteen. If this surprises you, you will misinterpret the results of almost every scientific study you hear about.

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4 Comments leave one →
  1. April 7, 2014 9:57 pm

    How do you do that math? I didn’t get anything close to 7%.

    0.8% of women age 40 who have a screening with a positive test have cancer.

  2. April 7, 2014 10:24 pm

    Your second statement is false. Let’s walk through the math. 1%*80%=0.8% of women who have a screening, both have a positive result and have cancer. (Conversely, 0.2% both have a negative result and have cancer.)

    Similarly, 99%*9.6%=9.504% both have a positive result and do not have cancer. (Conversely, 89.496% both have a negative result and do not have cancer.)

    Therefore 0.8%+9.504%=10.304% have a positive result. (Conversely, 0.2%+89.496%=89.696% have a negative result.)

    Therefore, of those who have a positive result, 0.8%/10.304%=7.76% (approximately) have cancer. (Of those who have a negative result, 0.2%/89.496%=0.223% (approximately) have cancer.)

    So those who get a positive result, now know they have a 7.76% chance of having cancer, when before they had a 1% chance of it. Those who get a negative result, now know they have a 0.223% chance of having cancer, when before they had a 1% chance of it.

    If it’s still confusing, read the article linked; it explains the reasoning better than I can.

  3. Steve Johnson permalink
    July 13, 2014 9:41 pm

    Here’s a trick I learned for solving these problems on an actuarial exam.

    Start with a total pool of women – say 100,000 to make it easy.

    99,000 do not have breast cancer. 9,504 (9.6% of these) test positive anyway. 89,496 (90.4%) test negative.

    1,000 do have breast cancer. 800 (80%) test positive. 200 test negative.

    The total population of women who tested positive is 800 + 9,504 = 10,304. You’re interested in the percentage of that 10,304 who are actually positive – 800. 800/10,304 ~= 0.0776.

    About 1 in 13.

    Now, send me your girlfriend (Yudkowski is “polyamorous” and gets fed women by members of his cult).

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