# Testing Bayesian Concepts in R: using the Poission-Gamma Conjugate Priors to compute the Posterior Distribution

• In this article, the Poisson-Gamma Conjugate Priors will be used to compute the Posterior Probabilities of the number of customer arrivals in a retail shop every 10 minutes time window (can be modeled by a Poisson process).
• First, a Γ(α,β) prior is chosen with α=4,β=0.4 (consistent with our belief that the mean number of customers arrived in the store in a 10-min time window is 10, with a standard deviation of 5) to model the unknown mean customer arrival rate λ variable, so that λΓ(4,0.4).
• Then a few trials of a random experiment simulating the customer arrival process are conducted to collect the data and update the prior belief about λ from the likelihood, which can be modeled as i.i.d. Poisson random variables, YiPois(λ),λΓ(α,β),i.
• The posterior probability distribution is also a Gamma distribution as shown in the figure below from the videos of professor Herbert Lee.
• Then the recursive Bayesian updates and the prior and posterior hyper-parameters and the means are updated as and when a new datapoint is received. Also, the frequentist’s MLE and 95% confidence interval are computed, along with the Bayesian 95% credible interval.
• The following animation shows the results of simulation of customer arrivals in 20 such time intervals (each of 10 mins), starting with the prior Γ(4,0.4).
• The left bottom barplot visualizes simulated # customers arrived in every 10 mins window.
• Every time a new datapoint is received (# customers arrived at the shop in the next 10 mins window), the prior belief is updated.
• The right bottom table represents the summary statistics. Prior and Posterior means respectively correspond to the previous and updated beliefs about the #customers arrived at the shop in a 10 mins time window.
• The next animation shows the same results starting with a vague prior Γ(ϵ,ϵ).