How does Bayesian estimation differ from frequentist?

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AcadiFi · Accepted Answer

Frequentist: parameters are fixed unknowns; data is random. Bayesian: parameters are random variables with probability distributions; data updates beliefs.

The fundamental formula is Bayes' rule: posterior proportional to likelihood * prior. Given prior p(theta) encoding beliefs before seeing data, and likelihood p(x|theta) from the model, the posterior is p(theta|x) = p(x|theta) * p(theta) / p(x).

Tobias Reinholtz at Tallinn Asset Management estimates the mean return of a hedge fund strategy. He has 36 months of returns with sample mean 0.82% and sample std 3.1%. Frequentist gives mu_hat = 0.82%, 95% CI from -0.23% to 1.87%.

Bayesian: Tobias's prior belief (from similar strategies) is that monthly alpha averages 0.3% with prior std 0.6%. The posterior for mu:
- Prior precision: 1/0.6^2 = 2.78
- Data precision: 36/3.1^2 = 3.75
- Posterior mean = (2.78 * 0.3 + 3.75 * 0.82) / (2.78 + 3.75) = 0.60%
- Posterior std = sqrt(1/(2.78+3.75)) = 0.39%

The Bayesian estimate shrinks toward the prior. With more data, posterior dominates prior. With sparse data, prior matters.

Bayesian advantages:
1. Incorporates prior information (expert views, historical patterns)
2. Full uncertainty quantification via posterior, not just point estimate
3. Natural for decision theory — expected utility over posterior
4. Handles small samples gracefully via shrinkage

Costs:
1. Prior specification can feel subjective
2. Computationally expensive when posterior lacks closed form (requires MCMC)
3. Results depend on prior in small samples

Black-Litterman is Bayesian portfolio optimization: market equilibrium as prior, investor views as data, posterior expected returns for Markowitz optimization.

How does Bayesian estimation differ from frequentist?

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