Bayes’ Theorem

The mathematical formula for updating beliefs in light of new evidence. Published posthumously in 1763 by Thomas Bayes, it answers the question: given that I observed B, how should I revise my belief about A?

The Formula

P(A | B) = P(B | A) · P(A) / P(B)

In words:

• P(A) = prior — your belief about A before seeing evidence B
• P(B | A) = likelihood — how probable the evidence is if A is true
• P(B) = marginal likelihood — the total probability of seeing the evidence
• P(A | B) = posterior — your updated belief about A after seeing B

Using the Theorem of Total Probability, this is often written as:

P(A | B) = P(B | A) · P(A) / [P(B | A) · P(A) + P(B | A’) · P(A’)]

The Clinical Test Example

The most striking illustration from Cameron’s notes:

• A disease affects 1 in 1,000 people (P(carrier) = 0.001)
• A test is 99% sensitive (P(positive | carrier) = 0.99)
• The test has a 5% false positive rate (P(positive | non-carrier) = 0.05)

A patient tests positive. What’s the probability they’re actually a carrier?

P(carrier | positive) = (0.99 × 0.001) / [(0.99 × 0.001) + (0.05 × 0.999)] = 0.00099 / 0.05094 = 1.94%

Despite a “95% accurate” test, a positive result means less than 2% chance of disease. Why? Because the base rate (0.1% prevalence) is so low that the 5% of healthy people who falsely test positive vastly outnumber the tiny fraction of carriers who correctly test positive.

“There is a very big difference between P(A | B) and P(B | A).” — Peter Cameron

Why Bayes’ Theorem Matters

The Base Rate Problem

The clinical test example reveals humanity’s deepest probabilistic blind spot: base rate neglect. People hear “95% accurate” and assume a positive result is 95% reliable. They anchor on the test’s accuracy and ignore the prior probability.

This is the same error pattern documented throughout behavioral-psychology:

• Doctors overestimate disease probability after a positive screening test
• Juries overweight DNA evidence without considering base rates
• Investors overreact to confirming evidence for their thesis
• Media consumers treat rare dramatic events as common threats

Bayesian Updating as a Way of Thinking

Beyond the formula, Bayes’ Theorem encodes a worldview:

1. Start with a prior — your best estimate before new information
2. Observe evidence — what actually happens
3. Update proportionally — strong evidence from a reliable source warrants large updates; weak or ambiguous evidence warrants small ones
4. The posterior becomes the new prior — and the cycle continues

This is the mathematical formalization of fallibilism: all beliefs are provisional (priors), subject to revision (updating), and never certain (posteriors are probabilities, not certainties).

Connection to Judgment and Decision-Making

Naval’s concept of judgment — “knowing the long-term consequences of your actions” — is implicitly Bayesian. Good judgment means:

• Having well-calibrated priors (experience, domain knowledge)
• Updating correctly when new information arrives (not over- or under-reacting)
• Acting on posterior probabilities rather than wishful thinking

Munger’s emphasis on base rates in investing is explicitly Bayesian: before evaluating any specific investment thesis, ask “what’s the base rate of success for this type of investment?” Most people skip this step.

Connection to High Agency

High agency requires the willingness to update beliefs — to abandon a plan when evidence shows it’s wrong, to revise your model of the world. The Bayesian framework gives this a precise structure: the strength of your update should be proportional to the strength of the evidence.

The “Attachment Trap” from the high agency essay — clinging to beliefs because you’ve invested in them — is precisely a failure of Bayesian updating. A Bayesian reasoner treats sunk costs as irrelevant to the posterior.

Munger’s Bayesian Instinct

Munger doesn’t use the word “Bayesian,” but his decision-making framework is deeply Bayesian:

• “What’s the base rate?” — Always start with the prior
• “What are the odds?” — Think in probabilities, not certainties
• Circle of competence (circle-of-competence) — Your priors are only reliable within your domain of genuine knowledge
• Inversion (inversion) — Consider the probability of failure, not just success

His “Two-Track Analysis” (rational factors + subconscious psychological influences) can be read as: compute the Bayesian posterior (Track 1), then check what psychological biases might be distorting your estimate (Track 2).

Connections

• probability-theory — The mathematical framework within which Bayes’ Theorem lives
• mental-models — Bayesian reasoning as one of Munger’s core models
• behavioral-psychology — Base rate neglect as a systematic cognitive error
• anchoring-bias — Anchoring on evidence (the likelihood) without adjusting for the prior
• judgment — Bayesian reasoning as the mathematical structure of good judgment
• fallibilism — All beliefs as provisional priors, subject to updating
• high-agency — Willingness to update beliefs in light of evidence
• inversion — Considering P(failure) as a Bayesian prior
• loss-aversion — Asymmetric weighting of outcomes distorts expected value calculations
• first-principles-thinking — Bayesian reasoning strips away assumptions and reasons from evidence

Sources

• source—notes-on-probability — Peter Cameron’s mathematical treatment, including the clinical test example
• source—poor-charlies-almanack — Munger’s emphasis on base rates and probabilistic thinking in decision-making