Near the end of week one of the 2025 NFL season, the Buffalo Bills found themselves down 15 points to the Baltimore Ravens late in the fourth quarter. As the Bills’ punter kicked the ball away, ESPN’s online win probability tracker shifted the Ravens’ chances of victory to a towering 98.6%.
With Baltimore up by multiple touchdowns, most sports fans saw a game that seemed basically cooked. Michael Titelbaum saw yet another example of Bayesian philosophy at work.
The philosophical framework, based on a formula discovered in the 18th century by the Reverend Thomas Bayes, couches knowledge and belief in terms of degrees of confidence that shift when new information enters the picture. In the case of the Bills-Ravens game, the situation before the Bills punted reflected the current high probability — i.e., degree of confidence — of a Ravens win. In Bayesian philosophy, that’s known as the “prior.” As new information gets added to the situation — like Ravens running back Derrick Henry subsequently fumbling the ball a few plays later, giving the Bills new life and a thrilling comeback win — it changes the probability, leading to a new “posterior.”
“The posterior becomes your next prior, and you just keep iterating this process over and over for your whole life,” says Titelbaum, a Vilas Distinguished Achievement Professor of Philosophy. “That’s the outline of how a Bayesian thinks about what’s going on.”
Titelbaum first encountered Bayesian philosophy in 2003 as a graduate student living in Oakland. This was about the same time that Billy Beane, the general manager of Major League Baseball’s Oakland A’s, was using “Moneyball” analytics — you guessed it, another form of Bayesianism — to retool the team’s roster and become more competitive.
But as he looked around, Titelbaum began to see Bayesian thinking not just in sports, but everywhere. He saw it in fields like science, medicine and economics, each of which depends to some degree on statistics and probability.
Over the course of the 20th century, scientists went from relying on determinism — the idea that things in the world can only occur in one possible way — to indeterminism, where scientists look at patterns and predict the probability that they’ll continue.
“When you’re doing that, as philosophers started realizing as early as the 1700s, you’re not really saying, ‘I’ve seen this pattern, therefore this prediction will definitely happen,’” Titelbaum explains. “You’re saying, ‘I’ve seen this pattern, therefore this is likely to happen.’”
Titelbaum says that the philosophical story about how science works became a story about how evidence makes certain conclusions probable — which is, of course, a key Bayesian tenet.
“You’ve got some hypothesis about how a particular system works. You think that hypothesis has a certain probability, and then you perform an experiment,” Titelbaum says, explaining the Bayesian apparatus. “When the data comes in from the experiment, how should you change your level of confidence in the hypothesis?”
Over the last decade, the legalization of sports gambling has only accelerated the rise of Bayesian thinking, as has the ups and downs of the stock market. But the legalization of sports gambling has also accelerated something else. Bayesian philosophy traffics in probabilities, and Titelbaum notes that people are notoriously bad at thinking of probabilities in terms of percentages. As a result, they often make mistakes when doing things like betting on sports, a flaw betting sites are only too happy to leverage. For example, an ESPN tracker might say a football team has a 60 percent chance of winning at a particular moment during the game, but the betting site is likely to register that chance in terms of odds (like +170). Most people can’t make that mental conversion, especially on the fly.
“There are all these studies that show that if you want people to draw conclusions from probabilities, they deal much better with probabilities expressed as something like 10 to one, or even better expressed as one in 1,000,” says Titelbaum.
Titelbaum notes that most of the highly ranked philosophy departments at American colleges and universities now have at least one person who understands Bayesianism, a major shift from several decades ago. Over the summer, UW–Madison’s Department of Philosophy hosted a Bayesian Boot Camp, drawing scholars from across the country.
Some of that pull is likely driven by our continued fascination with technology. The large language models that fuel artificial intelligence are dependent on Bayesian thinking. Titelbaum likes to blow people’s minds by pointing out that the spam filters on email inboxes also work on Bayesian principles, with priors constantly being updated by new information.
The temptation to take a degree of confidence in anything and mathematize it carries risks, however.
“The problem is, once these mathematical ways of thinking become popular, everyone starts to demand it, even when it’s not appropriate,” he says. “People want evidence, and by evidence, they mean data, and by data they mean numbers, and they mean statistics. And for some things, statistics aren’t appropriate. I spend a lot of my professional energy talking about when it’s appropriate to use these tools and when it’s not.”
On Oct. 29, as part of his department’s Philosophers at Work public lecture series, Titelbaum will explore our growing need to quantify everything using data and some of the drawbacks that can entail in terms of how we view success and value.
For now, he can only marvel as Bayesianism’s seemingly unstoppable momentum continues.
“It’s funny that something I was pursuing for purely philosophical reasons suddenly became relevant in daily life and everybody’s referencing it,” he says. “I was listening to my favorite movie podcast, and someone said, ‘You know, my priors going into this movie were so and so.’ What has happened is that Bayesianism has taken over the world in this unexpected way, where it just used to be this sort of obscure philosophical thing.”