Bayesian Statistics

Bayesian statistics treats unknown quantities, such as a strategy's true win rate or a stock's expected return, as uncertain values described by probability distributions. You start with a prior distribution reflecting what you believe before seeing the data, then update it with the data to get a posterior distribution. This approach is natural for traders, who constantly update views as new information arrives, and it is especially useful when data is limited or noisy, which describes most of finance.

The core idea#

posterior ∝ likelihood × prior

• Prior: your belief before seeing new data.
• Likelihood: how probable the data is under each possible value. See Maximum Likelihood.
• Posterior: your updated belief.

This is Bayes' theorem applied to whole distributions. See Bayes' Theorem.

A worked example: estimating a win rate#

Shrinkage#

Pulling noisy estimates toward a sensible central value is called shrinkage. It is one of the most useful ideas in quantitative finance:

Shrinkage reduces the impact of noise and usually improves out of sample performance.

Credible intervals#

A Bayesian 95% credible interval contains the true value with 95% probability, given the model and prior. This matches how most people intuitively interpret intervals, unlike frequentist confidence intervals. See Confidence Intervals.

Bayesian methods in trading#

Choosing priors#

• Informative priors: based on previous research, similar strategies or economic theory.
• Weak or uninformative priors: spread out, letting the data dominate.
• Sceptical priors: centred on "no edge", useful for strategy evaluation, since most ideas do not work.

Results should be checked for sensitivity: if conclusions change completely with a slightly different prior, the data is not very informative.

Frequentist vs Bayesian#

Both have strengths; many practitioners use whichever suits the problem.

Frequently asked questions#

What is Bayesian statistics?#

An approach that treats unknown quantities as probability distributions and updates prior beliefs with data to form posterior beliefs.

What is shrinkage in finance?#

Pulling noisy estimates, such as expected returns or covariances, toward a sensible central value to reduce the effect of noise.

Why is Bayesian thinking useful for traders?#

Because market data is limited and noisy, and Bayesian methods combine prior knowledge with new evidence and update naturally as results arrive.

Next, learn the shapes data can take in Probability Distributions Explained.