Testing Bayesian Concepts in R: using the Poission-Gamma Conjugate Priors to compute the Posterior Distribution
Sandipan Dey
11 August 2016
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 modelled by a Poisson process).
First, a \(\Gamma(\alpha,\beta)\) prior is chosen with \(\alpha=4,\; \beta=0.4\) (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 variable # customer arrivals \(\lambda\) in a 10-min time window, so that \(\lambda \sim \Gamma(4,0.4)\).
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 \(\lambda\) from the likelihood, which can be modeled as i.i.d. Poisson random variables, \(Y_i \sim Pois(\lambda),\; \lambda \sim \Gamma(\alpha, \beta),\; \forall{i}\).
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 \(\Gamma(4,0.4)\).
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 \(\Gamma(\epsilon,\epsilon)\).
