Bayesian inference

Bayesian inference treats parameters as random variables and combines prior beliefs with observed data via Bayes' rule. For finance, the Bayesian framework formalises something every PM does intuitively: starting with views, updating them as new evidence arrives. Black-Litterman portfolio construction is Bayesian; modern ML risk models are Bayesian; the credibility-weighted credit spreads used in actuarial work are Bayesian.

Bayes' rule for parameters

p(θ | x) = p(x | θ) p(θ) / p(x)
         ∝ likelihood × prior

• p(θ): prior — what you believed before seeing data.
• p(x | θ): likelihood — same object MLE maximises.
• p(θ | x): posterior — updated belief after data.
• p(x): marginal likelihood / evidence — a normalising constant for parameter inference, but central for model comparison.

Conjugate priors — the magic shortcut

For certain likelihood-prior pairings the posterior has the same functional form as the prior, with updated parameters. Closed-form, no MCMC required.

• Normal mean (variance known) + Normal prior → Normal posterior. The classic Bayesian-mean update.
• Normal variance + inverse-gamma prior → inverse-gamma posterior.
• Bernoulli/binomial + Beta prior → Beta posterior. The default-rate updating used in credit.
• Poisson + Gamma prior → Gamma posterior. The actuarial credibility model.

Normal-normal update

Observe X₁, ..., Xₙ ~ N(μ, σ²) with σ² known. Prior μ ~ N(μ₀, τ²). Posterior:

μ | x ~ N(μ_post, τ²_post)
μ_post = (μ₀/τ² + n X̄/σ²) / (1/τ² + n/σ²)
1/τ²_post = 1/τ² + n/σ²

The posterior mean is a precision-weighted average of the prior mean and the sample mean. Precisions add. As n grows the prior is washed out; as τ² → ∞ (flat prior) the posterior mean converges to the MLE.

MCMC — when no conjugate prior helps

Markov Chain Monte Carlo (Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo) draws samples from intractable posteriors. The modern workhorses are Stan, PyMC, NumPyro — all use HMC variants. For a quant: MCMC is overkill for routine problems but essential when you have a structural model with non-standard priors (e.g., hierarchical credit models).

Posterior predictive

p(x_new | x) = ∫ p(x_new | θ) p(θ | x) dθ

The right way to predict future data: integrate over the posterior, not just plug in θ̂. The predictive distribution is wider than the likelihood evaluated at θ̂, correctly reflecting parameter uncertainty. Bayesian VaR is wider than MLE-plug-in VaR for the same reason.

Black-Litterman in one slide

BL is Bayesian portfolio construction: start with an equilibrium prior π (CAPM-implied returns), specify subjective views Q with confidence Ω, and combine via Bayes' rule. The posterior mean Π* is the precision-weighted blend; the posterior covariance updates Σ. Plug into mean-variance optimisation and you get the BL portfolio. Module 8 of Portfolio Theory walks through every step.