Prior Distribution

Understanding prior distributions

Prior distributions are fundamental to Bayesian inference in portfolio allocation and quantitative trading. They provide a mathematical framework for incorporating:

• Historical market behavior
• Domain expertise
• Economic theory
• Model constraints

The prior distribution $p(\theta)$ represents uncertainty about parameters $\theta$ before observing data. When combined with new evidence through the likelihood function, it produces the posterior distribution.

Types of prior distributions

Informative priors

Informative priors encode specific beliefs about parameters:

• Market volatility following patterns
• Mean reversion tendencies
• Correlation structures between assets

For example, in options pricing, prior beliefs about volatility might follow:

$\sigma \sim \text{LogNormal}(\mu_0, \sigma_0^2)$

This encodes the belief that volatility is positive and right-skewed.

Non-informative priors

When little prior knowledge exists, non-informative priors aim for minimal impact on inference:

• Uniform distributions over reasonable ranges
• Jeffreys priors that are invariant to parameter transformations
• Reference priors maximizing expected information gain

Applications in financial modeling

Portfolio optimization

Prior distributions help regularize portfolio weights by encoding:

• Turnover constraints
• Position sizing limits
• Factor exposures

The optimization becomes:

$w^* = \arg\max_w \{\mathbb{E}[U(w^T r)] + \lambda \log p(w)\}$

where $p(w)$ is the prior on portfolio weights.

Risk modeling

Priors improve risk estimates by:

• Stabilizing covariance matrices
• Incorporating regime beliefs
• Constraining risk factor loadings

For example, a hierarchical prior on correlations might be:

$\rho_{ij} \sim \text{Beta}(\alpha, \beta)$ $\alpha, \beta \sim \text{Gamma}(k, \theta)$

Mathematical properties

The prior distribution must satisfy probability axioms:

1. Non-negativity: $p(\theta) \geq 0$
2. Integration to 1: $\int p(\theta)d\theta = 1$
3. Support matching parameter space

Common choices include:

• Normal distributions for location parameters
• Inverse-gamma for variance parameters
• Dirichlet for probability vectors

Practical considerations

Prior elicitation

Converting domain knowledge to distributions requires:

• Structured expert interviews
• Historical data analysis
• Sensitivity testing
• Cross-validation

Computational aspects

Prior choice affects:

• Sampling efficiency
• Convergence rates
• Numerical stability

Conjugate priors can simplify computation but may be less realistic.

Impact on inference

The influence of the prior depends on:

• Sample size
• Data informativeness
• Prior specificity

As more data arrives through Bayesian updating, the likelihood typically dominates the prior's effect.

The process combines prior $p(\theta)$ with likelihood $L(x|\theta)$ via Bayes' rule:

$p(\theta|x) \propto L(x|\theta)p(\theta)$

This updating mechanism provides a formal framework for learning from market data while incorporating domain expertise.