Bayesian Machine Learning for Trading

Bayesian methods provide a principled framework for incorporating uncertainty into trading decisions. Unlike traditional frequentist approaches that provide point estimates, Bayesian inference gives us full probability distributions over parameters and predictions, allowing for better risk management and decision-making under uncertainty.

This chapter covers PyMC fundamentals, Bayesian strategy evaluation, and stochastic volatility models – all essential tools for uncertainty-aware algorithmic trading.

Introduction to Bayesian Inference

Bayesian inference is based on Bayes’ theorem:

P(θ|data) = P(data|θ) × P(θ) / P(data)

Where:

• P(θ|data) is the posterior: what we learn about parameters θ after seeing the data
• P(data|θ) is the likelihood: how probable the data is given parameters θ
• P(θ) is the prior: our beliefs about θ before seeing the data
• P(data) is the evidence: normalizing constant

Why Bayesian Methods for Trading?

1. Uncertainty Quantification: Get probability distributions, not just point estimates
2. Incorporate Prior Knowledge: Use domain expertise through informative priors
3. Robust to Overfitting: Bayesian regularization naturally prevents overfitting
4. Adaptive Learning: Update beliefs as new data arrives
5. Better Risk Management: Understand full distribution of possible outcomes

Bayesian Inference Workflow

The following diagram shows the end-to-end Bayesian inference pipeline as implemented in Puffin’s puffin.models.bayesian and puffin.models.stochastic_vol modules.

flowchart TD
    classDef prior fill:#2d5016,stroke:#1a3a1a,color:#e8e0d4
    classDef likelihood fill:#1a3a5c,stroke:#0d2137,color:#e8e0d4
    classDef posterior fill:#6b2d5b,stroke:#4a1a4a,color:#e8e0d4
    classDef prediction fill:#8b4513,stroke:#5a2d0a,color:#e8e0d4
    classDef data fill:#4a4a2d,stroke:#33331a,color:#e8e0d4
    classDef decision fill:#2d4a4a,stroke:#1a3333,color:#e8e0d4

    A[Domain Knowledge]:::prior --> B[Define Priors P θ]:::prior
    C[Market Data]:::data --> D[Construct Likelihood P data θ]:::likelihood
    B --> E[MCMC Sampling PyMC]:::posterior
    D --> E
    E --> F[Posterior Distribution P θ data]:::posterior
    F --> G[Convergence Diagnostics]:::posterior
    G -->|r_hat ≈ 1.0| H[Parameter Estimates with Uncertainty]:::prediction
    G -->|Divergences| E
    H --> I[Predictions with Credible Intervals]:::prediction
    H --> J[Bayesian Sharpe Ratio]:::prediction
    I --> K[Trading Signals]:::decision
    J --> L[Strategy Comparison]:::decision
    K --> M[Portfolio Allocation]:::decision
    L --> M

    style A fill:#2d5016,stroke:#1a3a1a,color:#e8e0d4
    style C fill:#4a4a2d,stroke:#33331a,color:#e8e0d4

Chapter Contents

Sub-Page	Topics
PyMC Fundamentals	Bayesian inference intro, PyMC basics, Bayesian linear regression, factor models
Bayesian Sharpe Ratio	Bayesian Sharpe ratio, strategy comparison with uncertainty
Stochastic Volatility	Dynamic hedge ratios, stochastic vol models, practical trading examples

Key Implementations in Puffin

• BayesianLinearRegression: Factor models with uncertainty
• bayesian_sharpe(): Robust strategy evaluation
• compare_strategies_bayesian(): Principled strategy ranking
• BayesianPairsTrading: Dynamic hedge ratios for pairs trading
• StochasticVolatilityModel: Time-varying latent volatility
• estimate_volatility_regime(): Quick volatility regime detection

Summary

Bayesian methods provide powerful tools for trading:

1. Uncertainty Quantification: Full probability distributions enable better risk management
2. Adaptive Learning: Update beliefs as new data arrives
3. Robust Inference: Handle outliers and fat tails naturally
4. Principled Comparison: Compare strategies accounting for estimation uncertainty

Notebook: Run the examples interactively in ml_models.ipynb

Related Chapters

• Part 9: Time Series Models – Bayesian methods naturally extend time series models with uncertainty estimates
• Part 8: Linear Models – Linear regression provides the foundation for Bayesian linear factor models
• Part 24: Risk Management – Posterior distributions from Bayesian inference inform position sizing and risk controls

Source Code

Browse the implementation: puffin/models/

Further Reading

• Bayesian Methods for Hackers by Cameron Davidson-Pilon
• Statistical Rethinking by Richard McElreath
• PyMC Documentation
• ArviZ Documentation
• “Bayesian Methods in Finance” by Rachev et al.

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Table of contents

• PyMC Fundamentals
• Bayesian Sharpe Ratio
• Stochastic Volatility