Beta Distributions: A Cornerstone of Bayesian Calibration | by Maham Haroon | Oct, 2023

Exploring the Versatility of Beta Distributions in Bayesian Inference

Hi there!

Distributions may not seem like a complex concept at first glance, but they are incredibly powerful and fundamental in the world of data analysis and statistics. Think about it this way: if you were to gather 50 shirts in various sizes and colors, you would have created a color distribution, a size distribution, and perhaps even a “how much does this shirt annoy you” distribution (jokingly, of course). The point is that as long as you have a category to measure, there’s a distribution waiting to be explored.

So, what exactly is a distribution? It’s essentially a way to show how a category spreads across a scale of probabilities or likelihoods. You can figure this out either from the data you have or from what you know about a particular topic. You’ve probably heard of terms like the normal distribution, skewed distribution, long-tailed distribution, and so on — each of these describes how data points are shaped.

Today I wanted to touch on the Beta Distribution and specifically its application in Bayesian Calibration. Bayesian Calibration is an approach that updates Bayesian inference with new data to find the best-fitting values for a model’s parameters. It considers both the prior information available about these parameters and the likelihood of the observed data given those parameters.

Before we dive into Bayesian Calibration with the Beta Distribution, let’s cover some technical details. Once we have those basics down, we’ll explore the Bayesian Calibration with Beta Distributions with an intriguing scenario.

The beta distribution, denoted as Beta(α, β), is a probability distribution characterized by two parameters. Its probability density function (pdf) is expressed as follows:

In this equation, both α and β represent the hyperparameters, and it’s important to note that they must always be greater than 0. Additionally, for the…