Sunday, December 20, 2009

A Call to (Bayesian) Missionaries

Math is Tough

I need someone to show me the light.

I've tried, but I just can't seem to understand the far-reaching implications of Bayes' theorem. The more people I ask for help, the more I become convinced that 'Bayesian' is a synonym for 'inarticulate math-head'.

See, I want to understand why it matters. Why it's supposedly transformational.

Here's the example I have heard most frequently used to explain Bayes' theroem: Imagine there is a test to screen for a particular kind of cancer. When the test is administered to people who don't have the cancer, 99.5% of those screened will be correctly flagged as cancer-free (low false positive rate). When the test is administered to people who DO have the cancer, 99.9% of those screened will be correctly flagged as having the cancer (low false negative rate). In the general population, .01% of people have this cancer, so that would be about 30,000 people in the US!

Two questions: 1) Assuming the test is fairly cheap to administer, should this screening be done routinely for all people? 2) If I am flagged positive for having the cancer, what is the probability that I actually have the cancer?

The answer to question 2 is surprising, and that's why this example is used to promote Bayes' theorem. Using Bayes' theorem (I'll spare you the math) you find that if you're flagged as having the cancer, there is only about a 2% CORRECTION 16.7% chance that you actually have it. So the answer to question 1 is probably no, because getting a positive result still means you have a very low chance of actually having the cancer (instead the screening should probably only be done for people at high risk).

That seems like a pretty good example because it's easy to relate to and has a non-intuitive outcome. However, I think it must actually be a very poor example of the power of Bayes' theorem, because I can very easily work that same problem out with just standard probabilistic reasoning and get the same answer! You just divide the number of genuine positives (people who have the cancer who test positive for the cancer) by the number of total positives (false positives plus genuine positives) to get the likelihood that any particular positive result is a genuine positive result (the 2% result comes from the fact that the false positive rate is high relative to how rare the cancer is). It's simple, easy, and doesn't require Bayes' theorem.

But there are a lot of people who are excited about Bayes and his famous theorem, so there must be more to it than this example. So can someone please help me to understand?


  1. The importance of Bayes' theorem is that it also applies to probability densities and in the Bayesian scheme one can expresses one's state of knowledge of an unknown quantity by a probability distribution. The theorem shows how to update that distribution, in the light of new evidence, to a new distribution. Thus it is the completely general, completely rigorous mathematical description of how one should learn from noisy data.

  2. Thanks bookboy! I'm starting to understand better (see today's post).

    Maybe you can give me some history on why this insight is (I'm assuming) fairly recent? Bayes' theorem has been around for quite awhile, so why is it that it took so long to get to Popper's falsification approach to knowledge, and then to have Popper surpassed by Bayes? Do you follow what I'm asking and know something about the answer?