Bayes' theorem — updating beliefs with evidence

Two lines of algebra

You already own everything needed to derive Bayes' theorem. Quick reading reminder from A1-05: $P(A \mid B)$ is read "the probability of A given B" — the chance A is true once you know B happened. One new symbol: $\cap$ is read "and", so $P(A \cap B)$ is "the probability that A and B both happen" — the joint probability.

From A1-05, that joint probability factors two ways:

$P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$

In words: the probability of "A and B both" equals the probability of B, times the probability of A given B. It also equals the probability of A, times the probability of B given A. Two routes, same number.

Divide both sides by $P(B)$:

$P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}$

In words: the probability of A given B equals the probability of B given A, times the probability of A on its own, divided by the probability of B on its own. (The horizontal bar is a fraction: top divided by bottom.)

That's it — that is Bayes' theorem. It is the exchange rate between the two conditionals from A1-05's inversion trap: it converts "probability of B given A" into "probability of A given B". And $P(A)$ — the base rate, also called the prior — your probability for A before seeing any evidence — is what you pay at the exchange.

In practice $P(B)$ is expanded over the ways $B$ can happen. One more symbol: $\neg A$ is read "not A" — the event that A does not happen:

$P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B \mid A)\,P(A) + P(B \mid \neg A)\,P(\neg A)}$

In words: the bottom adds up the two ways B can occur — B happening when A is true, plus B happening when A is false. So evidence $B$ can come from the world where $A$ is true, or from the world where it isn't. The posterior — your updated probability of A after seeing the evidence — is just the share of all $B$-occurrences that come from the $A$-world.

The diagnostic example, in counts

Formulas hide intuition; counts reveal it. Take 10,000 people. The disease base rate is 1%, so 100 of them actually have it. Sensitivity — the share of real cases the test catches — is 99%. The false-positive rate — the share of healthy people who wrongly test positive anyway — is 5%:

$P(\text{disease} \mid \text{positive}) = \frac{99}{99 + 495} = \frac{99}{594} \approx 0.167$

In words: the probability of disease given a positive test is the number of true positives (99) divided by the number of all positives (99 + 495 = 594). The $\approx$ sign reads "approximately equals": about 0.167, i.e. 17%.

The false positives (495) swamp the true positives (99) — not because the test is bad, but because healthy people outnumber diseased ones 99 to 1. When the condition you're testing for is rare, even a good test yields mostly false alarms.

Your backtest passed. So what?

Now the version of this problem you will live inside for the rest of this curriculum. A backtest — a simulation of a trading strategy on historical data — plays the role of the medical test. Replace "disease" with "strategy has a real edge" and "positive test" with "backtest looks good":

• Prior: most strategy ideas have no edge. Be generous and say 5% of the ideas you'll ever test are genuinely good.
• Power — the chance the test detects a real effect when one exists: a good backtest protocol catches a real edge, say, 60% of the time (real edges are small and noisy — more in A2-04).
• False pass rate: a no-edge strategy passes anyway 5% of the time. (That 5% is the classic significance threshold — the error rate statistical convention tolerates — A2-05.)

$P(\text{edge} \mid \text{pass}) = \frac{0.60 \times 0.05}{0.60 \times 0.05 + 0.05 \times 0.95} = \frac{0.030}{0.0775} \approx 0.39$

In words: the probability of a real edge given a passing backtest. The top is the chance of "real edge AND pass": 0.60 × 0.05 = 0.030. The bottom is the chance of a pass from either world: passes from real edges (0.030) plus lucky passes from the 95% of no-edge strategies (0.05 × 0.95). Divide: 0.030 / 0.0775, approximately 0.39.

A passing backtest, under honest and even somewhat optimistic assumptions, means roughly a 39% chance the strategy is actually good. Less than a coin flip. The pass moved you from 5% to 39% — genuinely valuable evidence. But treating it as a verdict ("it works!") is the diagnostic fallacy — mistaking "probability of a pass given an edge" for "probability of an edge given a pass" — with money attached.

And it gets worse. If you tested 100 variations and kept the best-looking one, your effective false-pass rate is far above 5%, and the posterior collapses back toward the prior. That is the multiple-testing problem — try enough things and something will pass by pure luck. A2-05 treats it in full; Bayes is why it's lethal.

Beliefs are distributions, not verdicts

The deeper shift Bayes asks of you: stop treating claims as true/false flags waiting to be set. Start treating them as probabilities that move as evidence arrives. Before the backtest: 5%. After one clean pass: 39%. After a year of live trading in line with the backtest: higher still. After a live year that looks nothing like the backtest: back down.

Nothing is ever "proven" — you just accumulate evidence and update. Full Bayesian statistics builds an entire inference framework on this one theorem. There, priors and posteriors are placed over parameters — the unknown numbers inside a model — not just over events. McElreath's Statistical Rethinking is the classic on-ramp. For now, the discipline to internalize is: always ask what the base rate is before celebrating a positive signal.

Predict before the next lesson

Next lesson: the Law of Large Numbers and the Central Limit Theorem — the machinery behind "more data reveals the truth." Predict:

• You flip a fair coin 1,000 times. Roughly what range of head-counts would surprise you?
• A casino's edge on one roulette spin is about 5%, yet a single gambler can walk away a winner. Why is the casino nonetheless certain to profit over a million spins?
• Yesterday's Bayes posterior of 39% was based on one backtest. What kind of additional evidence would push it above 90% — and how much of it do you guess you'd need?

Note your answers. We check them against the math next.