Probability Wars
The long-running debate between frequentist and Bayesian statisticians over how to define probability—objectively as long-run frequencies or subjectively as degree of belief—has become an identity battle with strong emotions and even songs.
THE IDEA
THE “PROBABILITY WARS”
No question is so dry or esoteric that it’s immune to the forces of identity. Don’t believe me? Allow me to introduce you to the long-running debate between the frequentists and the Bayesians, two camps of statisticians who analyze data differently. At the root of the debate is a simple philosophical disagreement.
Frequentists define the probability of an event objectively, in terms of how frequently it would occur in a long series of trials. A frequentist would say that the probability of a coin flip coming up heads is one-half, because if you could flip the coin an infinite number of times, half of the outcomes would be heads.
Bayesianism is based around a theorem called Bayes’s rule, named after the Reverend Thomas Bayes, the eighteenth-century philosopher and statistician who first came up with it. Bayesians define the probability of an event subjectively, in terms of how confident someone is that an event will occur. Remember the exercises we did in chapter 6, in which we learned to quantify our confidence in a claim by thinking about what bets we would be willing to take on it? A Bayesian would call those “probabilities”; a frequentist would not.
You might expect these debates to play out exclusively in technical language buried in academic journal articles. But for decades, there was an annual Bayesian conference where attendees sang songs cheering for Bayesianism and booing frequentism. Here’s a sample stanza, meant to be sung to the tune of “The Battle Hymn of the Republic”:
Mine eyes have seen the glory of the Reverend Thomas Bayes He is stamping out frequentists and their incoherent ways . . . Glory, glory, probability! Glory, glory, subjectivity! His troops are marching on.16
Obviously, this song is tongue-in-cheek. But like all good observational humor, it’s based on a seed of truth. If you browse the statistics blogosphere, you’ll find Bayesians and frequentists accusing each other of irrational prejudice, complaining about fundamentalist frequentists, orthodox frequentists, anti-Bayesian prejudice, smug Bayesians, angry anti-Bayesians, Bayesian cheerleaders, and die-hard Bayesians. One statistician even renounced Bayesianism and wrote a blog post titled “Breathing Some Fresh Air Outside of the Bayesian Church.”17
Like many identity battles, the probability wars began with Bayesians feeling embattled back in the 1980s. They had to be careful not to mention the “B-word” very often, lest they be seen as troublemakers. At least one professor who favored Bayesian methods was pushed out of his department for his dissent. “We were always an oppressed minority, trying to get some recognition,”18 remembered Alan Gelfand, one of the early adopters of Bayesianism. Now the tables have turned. In the last fifteen years, Bayesianism has become popular, and frequentists are the ones feeling marginalized—so much so that one frequentist statistician titled her blog post “Frequentists in Exile.”19
The probability wars have even spilled out of academia and into the nerdier parts of the internet. In 2012, the webcomic XKCD published a strip on the difference between frequentist and Bayesian approaches that poked fun at the former.20 The reaction was so heated that one commenter joked, “Next time make it Israelis and Palestinians, it’ll be less controversial.”21
When to use it
When designing a clinical trial, choosing between frequentist and Bayesian approaches can affect trial design, sample size, and interpretation of results.
In business analytics, using Bayesian updating allows incorporating prior knowledge from previous studies or expert opinion into new data analysis.
In daily decision-making, understanding that probability can be subjective helps evaluate risk assessments and predictions from different sources.
In AI and machine learning, Bayesian methods are used for probabilistic models and uncertainty quantification, while frequentist methods are common in classical statistics.
FROM THEORY TO EVERYDAY LIFE
In practice
Coin Flip Probability
A frequentist defines the probability of heads as 1/2 based on long-run frequency, while a Bayesian defines it as a measure of confidence that might differ if prior knowledge exists.
XKCD Comic Controversy
In 2012, the webcomic XKCD published a strip highlighting the difference between frequentist and Bayesian approaches, sparking a heated online debate with comparisons to the Israeli–Palestinian conflict.
A MOMENT TO REFLECT
Put it to work.
How might your own identity bias influence your choice of statistical methods in your work or studies?
Can you think of a situation where you preferred a method because it aligned with your group identity rather than the problem's needs? How could you guard against that?
In what ways do the emotional reactions of statisticians to these debates parallel other identity-driven conflicts you have observed?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Scout Mindset (Julia Galef)