Creation of Credible set based on HPD

        The Bayesian approach is much more different from the usual inference procedure that we are used to. The first and the most important part is that here we consider some prior information about θ and based on our data, we want to improve the information of θ for a further purpose, called posterior information. So the θ is a random quality here. Hence the estimation procedure becomes different from the usual one and totally based on the posterior information about the θ, π(θ|x) which is the conditional distribution of θ given data.

Credible Set

For an interval estimation in Bayesian Inference, we find a Credible Interval or perhaps sometimes a credible set within which the θ lies with a certain confidence.
        An interval [l(x), u(x)] based on the observed values x is called a credible interval at confidence level 1 - α if
                                                    
Sometimes, a set C ∈ Θ is called a credible set at level 1 - α was given the sample point x if                    

                                                         

Why HPD

It is preferable to choose an interval or set where the probability distribution has a higher chance to occur, so it is preferable to choose a HPD(Higher Posterior Density) credible set.

Define,  

And then take,

 and,

                                                        

                                                            

Here we will try to visualize how a HPD credible set is created, for a famous example in Bayesian Analysis, the Beta Binomial example.

Please Note: To find the credible set we should need to calculate the necessary results for the example, which is skipped here, as we are keeping our focus on visualizing the process of finding the credible set.

Get some more details from Github: Link to the GitHub Repository