What Are the Odds? Can Bayes Help?

We hairless apes are notoriously bad at estimating probabilities. I plan to write more about other such “cognitive biases.”

For example, if you see several coin tosses coming up heads, do you think that the odds go up for the next toss to be tails? Assuming a fair coin, the odds for each toss are in fact independent and equal.

In the 1700s, English statistician, philosopher, and Presbyterian minister Thomas Bayes developed a theory of probability that was quite revolutionary. He never published it, but it was discovered and published after his death by Richard Price.

Pierre-Simon Laplace was a French mathematician and physicist who believed in a clockwork universe, where the entire future was determined by present conditions. I plan to write more about these ideas. Yet he became the biggest promoter of the Bayes probability theory. Laplace was aware that actual human measurements would never be perfect, so predictions about the future would always be probabilistic. And Bayes offered the best theory.

One practical application of Bayes’ theory is in testing for disease. Most people know that any medical test has a chance of false negatives, where the test misses a disease. And every such test has false positives, where you don’t have the disease, but the test says you do.

Most women get mammograms to test for breast cancer. About 10% of these tests give a false positive. This can be terrifying. But it is important to consider something called the base rate. For women aged 40-50, the base rate is just 1 in 69 of having breast cancer in a given year. For younger women, the base rate is much lower. This is why most countries suggest young women not get mammograms. Because the risk of harm from a false positive is bigger than the risk of breast cancer.

Bayes’ theorem works backwards from our intuitions and factors in the base rate. We are used to working forward from causes to effects. But the genius of Bayes is that it works backwards from effects to causes. When we see a potential cause or signal like a positive test, we jump to the conclusion that the bad effect will follow. But we need to remember that the bad effect may be very rare. That is the base rate.

A very low base rate can cancel out the effects of false positives. People spend way too much time worrying about things that have low base rates. In the case of breast cancer, the base rate goes up significantly with age. Which is why older women should get mammograms.

A similar situation holds with using turn signals. People decide right-of-way based on whether the other driver is signaling or not. But if the base rate of using a turn signal is too low, it becomes almost useless. The Society of Automotive Engineers (SAE) studied this problem and concluded that two million accidents occur each year in the U.S. because of this base rate failure to use turn signals. About twice the number caused by distracted driving or by drunk driving.

Bayes’ probability theory is also used in spam filters for email. Even more important, it is used for attribution of climate crisis disasters. Properly applied, Bayes can tease out how much a destructive weather event can be attributed to the climate crisis.

You might think that Bayes locks us into missing important dangers if the base rate is low. For example, before 9/11 the base rate for terror attacks with airplanes was near zero. But Bayes tells us to update the base rate upon learning new information. After the first plane hit the World Trade Center tower, we might rapidly increase the odds of it being a terrorist attack to one in three. And the second hit might increase the odds to
nearly 100%.

I have learned not to put too much math in my articles. But there are plenty of good articles explaining Bayes’ probability theory if you want to know more. I highly recommend the book The Signal and the Noise: Why So Many Predictions Fail – but Some Don’t by legendary forecaster Nate Silver. He includes an excellent section on Bayes.

Bayes also radically changed how we view reality. He showed us that probability is not just a property of “reality.” It is also a property of what we know. The difference between ontology and epistemology – and that leads to some very Big Questions!