Entropy Pooling in Python

Due to popular demand from developers, this package contains the Entropy Pooling implementation from the fortitudo.tech Python package with a more permissive BSD 3-Clause license.

This package contains only one function called ep and has minimal dependencies with just scipy. See the examples for how you can import and use the ep function.

You can explore the examples without local installations using Binder.

Installation instructions

Installation can be done via pip:

pip install entropy-pooling

Theory

Entropy Pooling is a powerful method for implementing subjective views and performing stress-tests for fully general Monte Carlo distributions. It was first introduced by Meucci (2008) and refined with sequential algorithms by Vorobets (2021).

You can loosely think about Entropy Pooling as a generalization of the Black-Litterman model without all the oversimplifying assumptions. Entropy Pooling operates directly on the next generation market representation defined by the simulation matrix $R\in \mathbb{R}^{S\times I}$ and associated joint scenario probability vector $p\in \mathbb{R}^{S}$.

For a quick introduction to Entropy Pooling intuition, watch this YouTube video. For a collection of Entropy Pooling resources, see this Substack post.

The original Entropy Pooling approach solves the minimum relative entropy problem

$$q=\underset{x}{\text{argmin}}\lbrace x^{T}\left(\ln x-\ln p\right)\rbrace$$

subject to linear constraints on the posterior probabilities

$$Gx\leq h \quad \text{and} \quad Ax=b.$$

The constraints matrices $A$ and $G$ contain functions of the Monte Carlo simulation $R$ that allow you to implement subjective views and stress-tests by changing the joint scenario probabilities from a prior probability vector $p$ to a posterior probability vector $q$.

A useful statistic when working with Entropy Pooling is the effective number of scenarios introduced by Meucci (2012).

For a causal Bayesian network overlay on top of Entropy Pooling, see Vorobets (2023).

Video walkthroughs

Video walkthroughs of the two notebook examples are available here and here. The videos give additional insights into Entropy Pooling theory and its sequential refinements. It is highly recommended to watch these videos to quickly increase your understanding.

Portfolio Construction and Risk Management Book

Entropy Pooling is a core part of the next generation investment framework that also utilizes fully general Monte Carlo distributions and CVaR analysis, see this YouTube video for an introduction. To get a pedagogical and deep presentation of all the possibilities Entropy Pooling offers, see the Portfolio Construction and Risk Management Book.