Understanding Custom Distributions in PyMC

PyMC is a powerful probabilistic programming library in Python that allows for Bayesian inference using Markov chain Monte Carlo (MCMC) methods. While PyMC provides a wide range of ready-to-use probability distributions, there are cases where a specific application requires a custom distribution. In this article, we'll delve into defining a custom distribution in PyMC, which involves specifying the probability density function (PDF) and any necessary transformations.

Prerequisites

Before diving into custom distributions, it's essential to have a working knowledge of basic probability theory, Bayesian statistics, and PyMC itself. Familiarity with the following concepts will help:

• Probability density functions (PDFs)
• Markov Chain Monte Carlo (MCMC) methods
• PyMC's build-in distributions and their properties

Why Create a Custom Distribution?

1. Modeling Complex Phenomena: Some applications demand specific distributional assumptions not covered by standard distributions.
2. Incorporating Domain Expertise: If domain-specific knowledge suggests a particular probabilistic behavior, a custom distribution can embody this.
3. Combining Distributions: Sometimes it’s necessary to create a distribution that combines properties of multiple other distributions.

Components of a Custom Distribution

Defining a custom distribution in PyMC typically involves:

1. Probability Density Function (PDF): Specify the PDF, which describes the relative likelihood for a random variable to take on a given value.
2. Log Probability Function: This function calculates the logarithm of the PDF needed for MCMC.
3. Transformations: Handling constrained parameters, such as ensuring a parameter stays positive.

Steps to Define a Custom Distribution in PyMC

Let's illustrate the process with an example of a simple custom distribution.

Example: Creating a Truncated Normal Distribution

Suppose we want a Normal distribution but truncated to only allow non-negative values.

1. Define the Distribution Class
   We'll start by subclassing `pm.Continuous` since this is a continuous distribution:
   1 - pm.Normal.dist(mu=mu, sigma=sigma).cdf(lower)

• Numerical Stability: Ensure that the log probability function is numerically stable, especially for extreme values.
• Test Thoroughly: Validate the custom distribution through simulation studies to confirm its accuracy.
• Compatibility with MCMC: The custom distribution should work smoothly within PyMC's sampling algorithms.