# Bayes Knew You Were Going to Read This Review

Everything Is Predictable

By Tom Chivers. Atria/One Signal Publishers. 2024.

Review by James Lynch

WHEN I TOOK MATH STATS, Bayes’ theorem took a scant four of the 800 pages of the textbook (published 1990). The instructor told us some researchers were finding interesting applications for it, but we weren’t going to learn about that.

So much has changed. In Everything is Predictable: How Bayesian Statistics Explain Our World, author Tom Chivers describes how Bayes’ theorem has grown from statistical afterthought discovered by a 17th century English clergyman to a cornerstone of rational thought.

Most actuaries know that Bayes’ theorem calculates a conditional probability based on a prior forecast and new facts. Given events A and B,

Chivers doesn’t dwell on the formula.

(He believes, like Stephen Hawking, that each equation in a book halves sales.) He demonstrates how the theorem works with a number of examples.

Here’s one: Suppose 1% of women have breast cancer. A mammogram correctly identifi es the cancer 80% of the time but fails to do so 20% of the time. And 10% of the time the test yields a false positive—it diagnoses a cancer that’s not there.[1]

If a woman tests positive, how likely is it she has cancer?

If 100,000 women were tested, 1,000 would have cancer. Th e test would
have found 800 of them. Of the 99,000 cancer-free women, 9,900 would test positive. So only 800/10,700, or 7.5%, of the women testing positive would have cancer. More than 90% of women testing positive would be needlessly alarmed.

Th e multitude of examples makes it harder to follow the math. I wish he had used fewer examples and at least once actually demonstrated how the formula operates: [Event A is having cancer, and Event B is testing positive, so P(B|A) = 0.8; P(A) = 0.01; P(B) = P(B|A) + P(B|not-A) = 0.80.01 + 0.10.99 = 0.107, QED.]

Th e failure to understand Bayes’ theorem can lead to public policy failures. In the early days of COVID-19, Chivers notes, the U.K. considered issuing
“immunity passports” to people who tested positive for a COVID antibody, the presumption being they had survived the disease and could no longer pass it along. Chivers shows that false positives on the antibody test would have meant two-thirds of the passport holders actually had never had COVID and so remained potential carriers.

To tell the story of Bayes’ theorem and its philosophical successors, Chivers, a science writer for the international news platform Semafor, meanders through the disciplines of mathematics, religion, phi-losophy, and science. He is a thorough, skeptical researcher, and he describes complex concepts clearly and concisely.

Some of the book will be familiar to actuaries: an explanation of simple probability and statistical principles and the roles of Bernoulli, deMoivre and others in developing them. He writes for a lay reader to understand but moves the narrative so smoothly that his retellings never bored me.

Bayes was born around 1700 to a wealthy family of nonconformists—meaning they eschewed Th e Book of Common Prayer that was required in all British religious services. He studied in Edinburgh and became a minister like his father, and a mathematics hobbyist, as well. He was a defender of Newton’s calculus. (Skeptics felt understanding the math led to understanding planetary motion, which was feared to be a portal to atheism.) Later he turned to the then-trendy topic of probability. His famous insight appears in “An Essay towards Solving a Problem in the Doctrine of Chance,” published around 1755.

Perhaps because nonconformists were out of favor, Bayes’ work fell into obscurity; he died in 1761. Pierre-Simon Laplace independently discovered Bayesian principles in 1774 but relinquished claim to the theorem once he learned Bayes had been there first.

Frequentism is what I was learning in my math stats class 35 years ago and what most people learn today. You state a hypothesis, then test it against data. Th e data can force you to reject the hypothesis, but you never accept it as true.

Bayesians state a prior preference, then test it against data. The data allow them to embrace the prior estimate or establish a new one.

To me, actuarial credibility theory is an extension of Bayesian reasoning. A new estimate of price is a weighted average of the prior prediction and a new prediction based on the latest data. But there are frequentists who claim credibility as their own.

The Bayesian/frequentist rivalry is fierce. Reading this book, I concluded there are three types of statisticians: Bayesians, frequentists, and those who avoid the fray.

The debate dominates the middle third of the book. Chivers gives a fair account-ing of both sides. He acknowledges his own preference, as any Bayesian would.

What writer would to turn away from the Bayesians of the late 1970s? They were a rollicking bunch who embraced the underdog spirit in songs like “There’s No Theorem Like Bayes’ Theorem” or in these lyrics to “The Battle Hymn of the Republic,” written at a Spanish retreat:

Mine eyes have seen the glory of the
Reverend Thomas Bayes,
He is stamping out frequentists and their
incoherent ways,
He has raised his mighty army at the
Hotel Las Fuentes,
His troops are marching on!

Bayesian truths have indeed marched on. The framework has become a useful model for how people think.

A person has a particular belief, and they hold it with some degree of cer-tainty—similar to how a Bayesian has a prior estimate with its own probability dis-tribution. If new data confirms the belief, the estimate remains the same, and the degree of uncertainty shrinks. If new data defies the belief, the estimate is updated, and the degree of uncertainty grows.

Or as Paul Samuelson put it: “Well, when events change, I change my mind. What do you do, sir?”[2]

It’s a powerful concept. It explains how two people can see the same thing differ-ently, like a picture of a dress that looks blue and black to some and white and gold to others. They looked at the picture with different priors.

It also explains why it can be so hard to persuade someone to change their beliefs, despite “the facts.” Their prior is too entrenched.

Chivers has fun with examples of Bayes in real life:

• In Wordle, each guess provides information for updating your prior guess.
• For a Necker cube, your prior estimate determines what direction it seems to point.

And then there’s Charlie Chaplin’s mask, an illusion by Richard Gregory
(available online at richardgregory.org. It’s a video that begins with the convex side of a Halloween mask on a spindle, which turns slowly around. Its concave side … looks convex, too. The mask is shaded to create new data that overwhelms what we know we should see.

The endgame here is the Bayesian brain. A significant number of social sci-entists believe the brain creates a model of the world, anticipates what it is going to experience, then receives signals from the senses and corrects course, following Bayesian principles.

Chivers writes: “What you experience is not the data from your senses, but your predictions—predictions constantly updated by information from the senses, yes, but the world you live in is the predic-tion, not the data.” Our consciousness—what we experience—is not the world. It is our Bayesian model of the world.

So you don’t really see this sentence. You see a prediction of what it will say based on all of the information you’ve read in your life prior to this moment. That prediction gets updated based on what your eyes tell your brain what it has seen. Of course, that’s close to how we describe the operation of a large language model.

Deep stuff, eh?

There are also Bayesian hypotheses concerning mental health. A person with schizophrenia might have weak prior beliefs: a weak model of the world. Data from their senses overwhelm that model, and the brain has to develop a new, bizarre hypothesis to explain them.

A depressed person could be holding strong negative priors, so strong that the things that cheer up most people—praise and success—don’t help. Psychedelic drugs, some suggest, break down the strong priors and let the depressed person develop a more positive model.

At this point, Chivers gets a bit skep-tical himself, but his tone throughout seems to welcome skepticism. He happily relates that he sees Bayes “everywhere I look”: emails, evolution, etc.

Of course, even his book is a Bayesian journey. Reading it can update your priors. After you have finished, you might not see Bayes everywhere, but you’ll see it more than you used to.

JAMES LYNCH, MAAA, FCAS, is a freelance writer.

[1] I suspect Chivers made these numbers up.
[2] “When the Facts Change, I Change My Mind. What Do You Do, Sir?” The “Quote Investigator” website credits Samuelson. Keynes expressed a similar sentiment but wasn’t so pithy.

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