Exploring the Versatility of Beta Distributions in Bayesian Inference

Hi there!

Distributions may not seem to be a posh concept at first glance, but they’re incredibly powerful and fundamental on the earth of knowledge evaluation and statistics. Give it some thought this manner: for those who were to collect 50 shirts in various sizes and colours, you’ll have created a color distribution, a size distribution, and even perhaps a “how much does this shirt annoy you” distribution (jokingly, after all). The purpose is that so long as you’ve a category to measure, there’s a distribution waiting to be explored.

So, what exactly is a distribution? It’s essentially a strategy to show how a category spreads across a scale of probabilities or likelihoods. You’ll be able to figure this out either from the info you’ve or from what you understand about a selected topic. You’ve probably heard of terms like the conventional distribution, skewed distribution, long-tailed distribution, and so forth — each of those describes how data points are shaped.

Today I wanted to the touch on the Beta Distribution and specifically its application in Bayesian Calibration. Bayesian Calibration is an approach that updates Bayesian inference with recent data to search out the best-fitting values for a model’s parameters. It considers each 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’ve got 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 characterised by two parameters. Its probability density function (pdf) is expressed as follows:

On this equation, each α and β represent the hyperparameters, and it’s necessary to notice that they need to at all times be greater than 0. Moreover, for the…