In a recent podcast, we talked with professional poker player Annie Duke about thinking in probabilities, something good poker players do all the time. At the poker table or in life, it’s really useful to think in probabilities versus absolutes based on all the information you have available to you. You can improve your decisions and get better outcomes. Probabilistic thinking leads you to ask yourself, how confident am I in this prediction? What information would impact this confidence?
Bayes’s theorem is an accessible way of integrating probability thinking into our lives. Thomas Bayes was an English minister in the 18th century, whose most famous work, “An Essay toward Solving a Problem in the Doctrine of Chances,” was brought to the attention of the Royal Society in 1763—two years after his death—by his friend Richard Price. The essay did not contain the theorem as we now know it, but had the seeds of the idea. It looked at how we should adjust our estimates of probabilities when we encounter new data that influence a situation. Later development by French scholar Pierre-Simon Laplace and others helped codify the theorem and develop it into a useful tool for thinking.
Knowing the exact math of probability calculations is not the key to understanding Bayesian thinking. More critical is your ability and desire to assign probabilities of truth and accuracy to anything you think you know, and then being willing to update those probabilities when new information comes in. Here is a short example, found in Investing: The Last Liberal Art, of how it works:
Let’s imagine that you and a friend have spent the afternoon playing your favorite board game, and now, at the end of the game, you are chatting about this and that. Something your friend says leads you to make a friendly wager: that with one roll of the die from the game, you will get a 6. Straight odds are one in six, a 16 percent probability. But then suppose your friend rolls the die, quickly covers it with her hand, and takes a peek. “I can tell you this much,” she says; “it’s an even number.” Now you have new information and your odds change dramatically to one in three, a 33 percent probability. While you are considering whether to change your bet, your friend teasingly adds: “And it’s not a 4.” With this additional bit of information, your odds have changed again, to one in two, a 50 percent probability. With this very simple example, you have performed a Bayesian analysis. Each new piece of information affected the original probability, and that is Bayesian [updating].
Both Nate Silver and Eliezer Yudkowsky have written about Bayes’s theorem in the context of medical testing, specifically mammograms. Imagine you live in a country with 100 million women under 40. Past trends have revealed that there is a 1.4% chance of a woman under 40 in this country getting breast cancer—so roughly 1.4 million women.
Mammograms will detect breast cancer 75% of the time. They will give out false positives—say a woman has breast cancer when she actually doesn’t—about 10% of the time. At first, you might focus just on the mammogram numbers and think that 75% success rate means that a positive is bad news. Let’s do the math.
If all the women under 40 get mammograms, then the false positive rate will give 10 million women under 40 the news that they have breast cancer. But because you know the first statistic, that only 1.4 women under 40 actually get breast cancer, you know that 8.6 million of the women who tested positive are not actually going to have breast cancer!
That’s a lot of needless worrying, which leads to a lot of needless medical care. In order to remedy this poor understanding and make better decisions about using mammograms, we absolutely must consider prior knowledge when we look at the results, and try to update our beliefs with that knowledge in mind.
Weigh the Evidence
Often we ignore prior information, simply called “priors” in Bayesian-speak. We can blame this habit in part on the availability heuristic—we focus on what’s readily available. In this case, we focus on the newest information and the bigger picture gets lost. We fail to adjust the probability of old information to reflect what we have learned.
The big idea behind Bayes’s theorem is that we must continuously update our probability estimates on an as-needed basis. In their book The Signal and the Noise, Nate Silver and Allen Lane give a contemporary example, reminding us that new information is often most useful when we put it in the larger context of what we already know:
Bayes’s theorem is an important reality check on our efforts to forecast the future. How, for instance, should we reconcile a large body of theory and evidence predicting global warming with the fact that there has been no warming trend over the last decade or so? Skeptics react with glee, while true believers dismiss the new information.
A better response is to use Bayes’s theorem: the lack of recent warming is evidence against recent global warming predictions, but it is weak evidence. This is because there is enough variability in global temperatures to make such an outcome unsurprising. The new information should reduce our confidence in our models of global warming—but only a little.
The same approach can be used in anything from an economic forecast to a hand of poker, and while Bayes’s theorem can be a formal affair, Bayesian reasoning also works as a rule of thumb. We tend to either dismiss new evidence, or embrace it as though nothing else matters. Bayesians try to weigh both the old hypothesis and the new evidence in a sensible way.
Limitations of the Bayesian
Don’t walk away thinking the Bayesian approach will enable you to predict everything! In addition to seeing the the world as an ever-shifting array of probabilities, we must also remember the limitations of inductive reasoning. A high probability of something being true is not the same as saying it is true. A great example of this is from Bertrand Russell’s The Problems of Philosophy:
A horse which has been often driven along a certain road resists the attempt to drive him in a different direction. Domestic animals expect food when they see the person who usually feeds them. We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.
In the final analysis, though, picking up Bayesian reasoning can truly change your life, as observed in this Big Think video by Julia Galef of the Center for Applied Rationality:
After you’ve been steeped in Bayes’s rule for a little while, it starts to produce some fundamental changes to your thinking. For example, you become much more aware that your beliefs are grayscale. They’re not black and white and that you have levels of confidence in your beliefs about how the world works that are less than 100 percent but greater than zero percent and even more importantly as you go through the world and encounter new ideas and new evidence, that level of confidence fluctuates, as you encounter evidence for and against your beliefs.
So be okay with uncertainty, and use it to your advantage. Instead of holding on to outdated beliefs by rejecting new information, take in what comes your way through a system of evaluating probabilities.
Bayes’s Theorem is part of the Farnam Street latticework of mental models. Still Curious? Read Bayes and Deadweight: Using Statistics to Eject the Deadweight From Your Life next.