Defining a custom PyMC distribution

Defining Custom Distributions in PyMC

PyMC is a powerful probabilistic programming library for Bayesian statistical modeling. It offers a rich collection of pre-defined distributions, but sometimes, your model requires a distribution that’s not readily available. This is where defining custom distributions comes in handy.

The Power of Custom Distributions

• Model Flexibility: Tailor your models to specific problem domains, capturing nuanced relationships and complex dependencies.
• Domain Expertise: Incorporate expert knowledge and constraints directly into your model through custom distributions.
• Increased Accuracy: By using distributions that better represent the underlying data, you can achieve more accurate inference results.

Building Custom Distributions: A Step-by-Step Guide

1. The `Distribution` Class

PyMC distributions inherit from the Distribution class, providing a framework for defining their behavior. Let’s create a simple example: a custom uniform distribution on the interval [a, b].

 from pymc import Model, Uniform, sample import numpy as np class CustomUniform(Distribution): def __init__(self, a, b, *args, **kwargs): super().__init__(*args, **kwargs) self.a = a self.b = b def logp(self, value): if self.a <= value <= self.b: return 0.0 else: return -np.inf 

2. Defining Key Methods

• __init__: Initializes the distribution with its parameters (a and b in this case).
• logp: Implements the log probability density function (PDF). This method defines the likelihood of observing a given value.

3. Integrating with PyMC

To use your custom distribution in a PyMC model, you'll need to create an instance of it and add it as a random variable.

 with Model() as model: x = CustomUniform("x", a=0, b=10) trace = sample(2000, tune=1000) 

4. Visualization and Inference

You can now visualize the posterior distribution of x using PyMC's plotting tools or perform inference using techniques like MCMC sampling.

Example: A Custom Beta Distribution

Let's create a more complex example: a custom Beta distribution with an adjustable parameter for skewness.

 from pymc import Model, Normal, sample import numpy as np class SkewedBeta(Distribution): def __init__(self, alpha, beta, skewness, *args, **kwargs): super().__init__(*args, **kwargs) self.alpha = alpha self.beta = beta self.skewness = skewness def logp(self, value): if 0 <= value <= 1: return (self.alpha - 1) * np.log(value) + (self.beta - 1) * np.log(1 - value) \ + self.skewness * (value - 0.5) else: return -np.inf with Model() as model: alpha = Normal("alpha", mu=2, sigma=1) beta = Normal("beta", mu=2, sigma=1) skewness = Normal("skewness", mu=0, sigma=0.5) x = SkewedBeta("x", alpha=alpha, beta=beta, skewness=skewness) trace = sample(2000, tune=1000) 

By manipulating the skewness parameter, you can control the shape of the Beta distribution.

Conclusion

Defining custom distributions empowers you to build more tailored and accurate Bayesian models. By following the guidelines and examples presented in this article, you can easily extend PyMC's functionality and tackle a wider range of statistical problems.