Bayes' Theorem

Definition and Mathematical Foundation

A formula relating conditional probabilities: \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \). It provides a way to update prior beliefs given new evidence.

The theorem provides the mathematical foundation for updating beliefs in light of new evidence. It connects the prior probability \( P(A) \), the likelihood \( P(B|A) \), and the marginal \( P(B) \) to produce the posterior \( P(A|B) \). The denominator \( P(B) \) acts as a normalizing constant ensuring the posterior is a valid probability distribution.

Mathematical Details

For discrete hypotheses \( H_1, H_2, \ldots, H_n \) forming a partition:

\( P(H_i|D) = \frac{P(D|H_i)P(H_i)}{\sum_j P(D|H_j)P(H_j)} \)

For continuous parameters \( \theta \) with prior density \( \pi(\theta) \) and likelihood \( L(\theta|x) \):

\( \pi(\theta|x) = \frac{L(\theta|x)\pi(\theta)}{\int L(\theta|x)\pi(\theta)d\theta} \)

Example

A trading signal fires 70% of the time when the stock goes up, and 30% when it goes down. If stocks go up 55% of the time, what is the probability the stock goes up given the signal fired?

\( P(\text{Up}|\text{Signal}) = \frac{0.70 \times 0.55}{0.70 \times 0.55 + 0.30 \times 0.45} = \frac{0.385}{0.520} \approx 74\% \)

Application in Quantitative Finance

Foundational in quant interviews for probability brainteasers, Bayesian inference in trading models, and updating beliefs about market regimes given new data.

In quantitative trading, Bayesian methods are used for regime detection (updating the probability of bull vs. bear market as returns arrive), signal combination (combining multiple alpha signals with different reliabilities), and portfolio optimization (Black-Litterman model). The ability to formally update beliefs with data is what separates systematic from ad-hoc trading.

Interview Relevance

Bayes' theorem problems are among the most common in quant interviews. Expect variations of the classic disease testing problem, urn problems with updating, and multi-stage information revelation. The key skill is correctly identifying the prior, likelihood, and evidence, then computing the posterior without algebraic errors.