(Note: The author now considers this explanation obsoleted by the Bayes' Rule Guide.)

[Editor’s Note: This is an abridgement of the original version of this essay, which contained many interactive elements.]

Your friends and colleagues are talking about something called “Bayes’s Theorem” or “Bayes’s Rule,” or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a web page about Bayes’s Theorem and . . .

It’s this equation. That’s all. Just one equation. The page you found gives a definition of it, but it doesn’t say what it is, or why it’s useful, or why your friends would be interested in it. It looks like this random statistics thing.

Why does a mathematical concept generate this strange enthusiasm in its students? What is the so-called Bayesian Revolution now sweeping through the sciences, which claims to subsume even the experimental method itself as a special case? What is the secret that the adherents of Bayes know? What is the light that they have seen?

Soon you will know. Soon you will be one of us.

While there are a few existing online explanations of Bayes’s Theorem, my experience with trying to introduce people to Bayesian reasoning is that the existing online explanations are too abstract. Bayesian reasoning is very counterintuitive. People do not employ Bayesian reasoning intuitively, find it very difficult to learn Bayesian reasoning when tutored, and rapidly forget Bayesian methods once the tutoring is over. This holds equally true for novice students and highly trained professionals in a field. Bayesian reasoning is apparently one of those things which, like quantum mechanics or the Wason Selection Test, is inherently difficult for humans to grasp with our built-in mental faculties.

Or so they claim. Here you will find an attempt to offer an intuitive explanation of Bayesian reasoning—an excruciatingly gentle introduction that invokes all the human ways of grasping numbers, from natural frequencies to spatial visualization. The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else). When you are finished reading this, you will see Bayesian problems in your dreams.

And let’s begin.

Here’s a story problem about a situation that doctors often encounter:

1% of women at age forty

...

The Laws Governing Belief