How Bayesian Principles Help Us Make Better Predictions

This article is an excerpt from the Shortform book guide to "The Signal and the Noise" by Nate Silver. Shortform has the world's best summaries and analyses of books you should be reading.

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What are the principles of Bayes’ Theorem? How can they help you calculate the probability of an event?

Bayes’ Theorem suggests that you make better predictions when you consider the prior likelihood of an event and update your predictions in response to the latest evidence. Nate Silver discusses how the theorem encourages you to think while making predictions.

Let’s look at the two Bayesian principles that can help you think better.

The Principles of Bayesian Statistics

Bayes’ Theorem—named for Thomas Bayes, the English minister and mathematician who first articulated it—posits that you can calculate the probability of event A with respect to a specific piece of evidence B. He explains how to make this calculation and shares two Bayesian principles that can

To calculate the probability of event A with respect to evidence B, Silver explains, you need to know (or estimate) three things:

The prior probability of event A, regardless of whether you discover evidence B—mathematically written as P(A)
The probability of observing evidence B if event A occurs—written as P(B|A)
The probability of observing evidence B if event A doesn’t occur—written as P(B|not A)

Bayes’ Theorem uses these values to calculate the probability of A given B—P(A|B)—as follows:

(Shortform note: This formula may look complicated, but in less mathematical terms, what it’s calculating is [the probability that you observe B and A is true] divided by [the probability that you observe B at all whether or not A is true—or P(B)]. In fact, Silver’s version of the formula (as written above) is a very common special case used when you don’t directly know P(B); that lengthy denominator is actually just a way to calculate P(B) using the information we’ve listed above.)

Principle #1: Consider the Prior Probability

To illustrate how Bayes’ Theorem works in practice, imagine that a stranger walks up to you on the street and correctly guesses your full name and date of birth. What are the chances that this person is psychic? Say that you estimate a 90% chance that if this person is psychic, they’d successfully detect this information, whereas you estimate that a non-psychic person has only a 5% chance of doing the same (perhaps they know about you through a mutual friend). On the face of it, these numbers seem to suggest a pretty high chance (90% versus 5%) that you just met a psychic.

But Bayes’ Theorem reminds us that prior probabilities are just as important as the evidence in front of us. Say that before you met this stranger, you would’ve estimated a one in 1,000 chance that any given person could be psychic. That leaves us with the following values:

P(A|B) is the chance that a stranger is psychic given that they’ve correctly guessed your full name and date of birth. This is what you want to calculate.
P(A) is the chance that any random stranger is psychic. We set this at one in 1,000, or 0.001.
P(B|A) is the chance that a psychic could correctly guess your name and date of birth. We set this at 90%, or 0.9.
P(B|not A) is the chance that a non-psychic could correctly guess the same information. We set this at 5%, or 0.05.

Bayes’ Theorem yields the following calculation:

P(A|B)= 0.001 x 0.9/0.001 x 0.9 + 0.05(1-0.001) = .0009/.05085 = 0.017699

That’s an approximately 1.77% chance that the stranger is psychic based on current evidence. In other words, despite the comparatively high chances that a psychic stranger could detect your personal information while a non-psychic stranger couldn’t, the extremely low prior chance of any stranger being psychic means that even in these unusual circumstances, it’s quite unlikely you’re dealing with a psychic.

How Bayesian Principles Help Us Make Better Predictions