Bayes' Theorem
Bayes' Theorem is a mathematical formula for updating the probability of a hypothesis based on new evidence, integrating prior knowledge to refine predictions.
THE IDEA
Bayes’ theorem relates the probability of a hypothesis after observing evidence to its prior probability and the likelihood of that evidence under the hypothesis.
For evidence B with positive probability, the relation is P(A|B) = P(B|A) × P(A) / P(B). P(A) is the prior probability, P(B|A) is the likelihood, and P(A|B) is the posterior probability. The denominator accounts for how often the evidence occurs overall, including under alternatives to A.
The theorem is named after Reverend Thomas Bayes (1701–1761). His Proposition 9 used evidence to calculate limits on an unknown parameter and appeared in An Essay towards solving a Problem in the Doctrine of Chances (1763).
Pierre-Simon Laplace independently developed conditional-probability methods, first in 1774 and later in his 1812 Théorie analytique des probabilités.
Bayes' theorem is to the theory of probability what the Pythagorean Theorem is to geometry. —Sir Harold Jeffreys
When to use it
Medical diagnosis: interpret test results in light of disease base rates and test accuracy.
Spam filtering: classifying emails based on word frequencies and prior spam prevalence.
Evaluating DNA evidence: distinguish how likely the evidence is under competing hypotheses from the probability assigned to a hypothesis after considering the evidence.
Scientific hypothesis testing: updating the plausibility of a theory as new experimental data becomes available.
FROM THEORY TO EVERYDAY LIFE
In practice
Cancer Screening and Age
To update cancer probability using age, start with a prior prevalence and compare how common that age group is among people with and without the cancer. Alternatively, use prevalence measured directly for the age group. The probability of cancer given age is the result being estimated, not the likelihood to multiply by the prior.
Spam Email Filter
An email contains the word 'free'. A Bayesian filter combines the prior spam rate with how often that word appears in spam and non-spam messages. Both groups matter when estimating the probability of spam after observing the word.
A MOMENT TO REFLECT
Put it to work.
How can you apply Bayes' Theorem to update a belief you currently hold when new evidence arrives?
Think of a recent decision where you ignored relevant prior probabilities. How might Bayes' Theorem have changed your conclusion?
What prior assumptions are you implicitly making in a current professional or personal evaluation, and how could you formally incorporate evidence?
Often confused with
References
- An Essay towards solving a Problem in the Doctrine of Chances — Thomas Bayes (1763)
- Mémoire sur la probabilité des causes par les événements — Pierre-Simon Laplace (1774)
- Théorie analytique des probabilités — Pierre-Simon Laplace (1812)
CONTINUE THE CONVERSATION
Source texts
- Wisdom Theory 100 Mental Models (2020) (Wisdom Theory)