# 1. INTRODUCTION

Bayesian statistics is a way of reasoning about uncertainty in a systematic and mathematically rigorous way. It is named after the 18th-century mathematician and philosopher Thomas Bayes, who developed a theorem that provides a way to update our beliefs about a hypothesis as we collect new data.

In Bayesian statistics, we start with a prior probability distribution, which represents our beliefs about a hypothesis before we have seen any data. As we collect new data, we update this prior distribution using Bayes’ theorem to obtain a posterior probability distribution, which represents our beliefs about the hypothesis after taking into account the data we have observed.

# 2. EXAMPLE

To give an example, suppose we are interested in estimating the proportion of voters who will vote for a certain candidate in an upcoming election.

We start with a prior belief that the proportion of voters who will vote for the candidate is uniformly distributed between 0 and 1. As we collect new data, such as the results of a poll, we can update this prior distribution using Bayes’ theorem to obtain a posterior distribution that takes into account the new information we have.

In Bayesian statistics, the posterior distribution is often used to make predictions or decisions. For example, we might use the posterior distribution to estimate the probability that a certain event will occur, or to decide whether to accept or reject a hypothesis.

# 3. BETA DISTRIBUTION

One common probability distribution used in Bayesian statistics is the beta distribution. It is a continuous probability distribution that is defined on the interval [0,1], which makes it particularly well-suited for modeling proportions or probabilities.

The beta distribution has two parameters, often denoted by alpha and beta, which control the shape of the distribution.

The beta distribution is a conjugate prior for the binomial distribution, which means that if we assume