The Bayesian Trap

David Smith links to a video describing an application of Bayes’s Theorem and gives the example of medical tests:

If you get a blood test to diagnose a rare disease, and the test (which is very accurate) comes back positive, what’s the chance you have the disease? Well if “rare” means only 1 in a thousand people have the disease, and “very accurate” means the test returns the correct result 99% of the time, the answer is … just 9%. There’s less than a 1 in 10 chance you actually have the disease (which is why doctor will likely have you tested a second time).

Now that result might seem surprising, but it makes sense if you apply Bayes Theorem. (A simple way to think of it is that in a population of 1000 people, 10 people will have a positive test result, plus the one who actually has the disease. One in eleven of the positive results, or 9%, actually detect a true disease.)

This goes to sensitivity/recall (in the medical field, they call it sensitivity; in the documents world and in the Microsoft ML space, they call it recall):  True positives / (True positives + False negatives).  Supposing a million people, 1000 will have the disease.  Of those 1000, we expect the test to find 990 (99%).  Of the 999,000 people who don’t have the disease, we expect the test to produce 9990 false negatives (1%).  990 / (990 + 9990) = 9%.

Related Posts

Non-Linear Classifiers with Support Vector Machines

Rahul Khanna continues a series on support vector machines: In this blog post, we will look at a detailed explanation of how to use SVM for complex decision boundaries and build Non-Linear Classifiers using SVM. The primary method for doing this is by using Kernels. In linear SVM we find margin maximizing hyperplane with features […]

Read More

Vectors for Programmers

John Mount has a couple of videos available: We have just released two new free video lectures on vectors from a programmer’s point of view. I am experimenting with what ideas do programmers find interesting about vectors, what concepts do they consider safe starting points, and how to condense and present the material. Click through […]

Read More

Categories