Bayes theorem is the probability rule economists and statisticians use to update beliefs when new evidence arrives.
$$$$
The formula
$$ P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} $$
Read from left to right:
- (P(A)) is the prior belief about event or hypothesis (A),
- (P(B \mid A)) is the likelihood of observing evidence (B) if (A) is true,
- (P(A \mid B)) is the posterior belief after seeing the evidence.
Why it matters in economics
Economic decisions are often made under uncertainty. Agents revise beliefs about inflation, productivity, default risk, or policy credibility as data arrive. Bayes theorem gives the formal rule for doing that revision consistently.
Practical interpretation
If a central bank receives new inflation data, it should not throw away everything it previously believed. It should combine prior beliefs with the new signal, weighing how informative that signal is. Bayesian forecasting and modern macro estimation are built on that logic.
Related Terms
Knowledge Check
### What does Bayes theorem help you do?
- [x] Update a prior belief using new evidence
- [ ] Eliminate uncertainty from all data
- [ ] Measure GDP directly
- [ ] Replace probability with certainty
> **Explanation:** Bayes theorem combines existing beliefs with new information to form a revised probability.
### In Bayes theorem, what is the prior?
- [x] The belief held before seeing the new evidence
- [ ] The final revised probability
- [ ] The sample average
- [ ] The forecasting error only
> **Explanation:** The prior is the starting belief before the latest evidence is incorporated.
### What is the posterior probability?
- [x] The updated belief after combining the prior with the likelihood of the evidence
- [ ] The probability before any data are observed
- [ ] The same thing as the denominator only
- [ ] The probability of never revising beliefs
> **Explanation:** The posterior is the end result of Bayesian updating.