Bayesian ML – Dynamic Sharpe Ratios and Pairs Trading

In this chapter, we will introduce Bayesian approaches to machine learning (ML) and how their different perspective on uncertainty adds value when developing and evaluating trading strategies.

Bayesian statistics allows us to quantify uncertainty about future events and refine our estimates in a principled way as new information arrives. This dynamic approach adapts well to the evolving nature of financial markets. It is particularly useful when there are fewer relevant data and we require methods that systematically integrate prior knowledge or assumptions.

We will see that Bayesian approaches to machine learning allow for richer insights into the uncertainty around statistical metrics, parameter estimates, and predictions. The applications range from more granular risk management to dynamic updates of predictive models that incorporate changes in the market environment. The Black-Litterman approach to asset...

How Bayesian machine learning works

Classical statistics is said to follow the frequentist approach because it interprets probability as the relative frequency of an event over the long run, that is, after observing a large number of trials. In the context of probabilities, an event is a combination of one or more elementary outcomes of an experiment, such as any of six equal results in rolls of two dice or an asset price dropping by 10 percent or more on a given day).

Bayesian statistics, in contrast, views probability as a measure of the confidence or belief in the occurrence of an event. The Bayesian perspective, thus, leaves more room for subjective views and differences in opinions than the frequentist interpretation. This difference is most striking for events that do not happen often enough to arrive at an objective measure of long-term frequency.

Put differently, frequentist statistics assumes that data is a random sample from a population and aims to identify the...

Probabilistic programming with PyMC3

Probabilistic programming provides a language to describe and fit probability distributions so that we can design, encode, and automatically estimate and evaluate complex models. It aims to abstract away some of the computational and analytical complexity to allow us to focus on the conceptually more straightforward and intuitive aspects of Bayesian reasoning and inference.

The field has become quite dynamic since new languages emerged after Uber open sourced Pyro (based on PyTorch). Google, more recently, added a probability module to TensorFlow.

As a result, the practical relevance and use of Bayesian methods in ML will likely increase to generate insights into uncertainty and, in particular, for use cases that require transparent rather than black-box models.

In this section, we will introduce the popular PyMC3 library, which implements advanced MCMC sampling and variational inference for ML models using Python. Together with Stan...

Bayesian ML for trading

Now that we are familiar with the Bayesian approach to ML and probabilistic programming with PyMC3, let's explore a few relevant trading-related applications, namely:

• Modeling the Sharpe ratio as a probabilistic model for more insightful performance comparison
• Computing pairs trading hedge ratios using Bayesian linear regression
• Analyzing linear time series models from a Bayesian perspective

Thomas Wiecki, one of the main PyMC3 authors who also leads Data Science at Quantopian, has created several examples that the following sections follow and build on. The PyMC3 documentation has many additional tutorials (see GitHub for links).

Bayesian Sharpe ratio for performance comparison

In this section, we will illustrate:

• How to define the Sharpe Ratio (SR) as a probabilistic model using PyMC3
• How to compare its posterior distributions for different return series

The Bayesian estimation for two...

Summary

In this chapter, we explored Bayesian approaches to machine learning. We saw that they have several advantages, including the ability to encode prior knowledge or opinions, deeper insights into the uncertainty surrounding model estimates and predictions, and suitability for online learning, where each training sample incrementally impacts the model's prediction.

We learned to apply the Bayesian workflow from model specification to estimation, diagnostics, and prediction using PyMC3 and explored several relevant applications. We will encounter more Bayesian models in Chapter 14, Text Data for Trading – Sentiment Analysis, where we'll discuss natural language processing and topic modeling, and in Chapter 20, Autoencoders for Conditional Risk Factors and Asset Pricing, where we'll introduce variational autoencoders.

The next chapter introduces nonlinear, tree-based models, namely decision trees, and shows how to combine multiple models into an ensemble...