*(Sandipan Dey, **11 August 2016)*

- In this article, the Exponential-Gamma Conjugate Priors will be used to compute the Posterior values for the
*customer arrival rate*in a retail shop (Inter-arrival times can be best modeled by an*exponential distribution*).

- First, a
**Γ(α,β) prior**is chosen with**α**=4,**β**=0.4 (consistent with our belief that the mean customer arrival rate in the store is 1/3, so that mean arrival time is 3 mins, with a standard deviation of arrival rate of 1/9) to model the unknown mean*customer arrival rate***λ**variable, so that**λ ∼ Γ(9,27)**.

- 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 (the ) can be modeled as an*exponential*random variable,**Y ∼ exp(λ)**,**λ ∼ Γ(α,β)**.

- The posterior probability distribution is also a
*Gamma*distribution as shown in the figure below from the videos of professor Herbert Lee.

- Then the
and the*recursive Bayesian updates**prior*and*posterior hyper-parameters*and the means are updated as and when a new datapoint is received. Also, the*frequentist’s*and**MLE***95% confidence interval*are computed, along with the Bayesian*95% credible interval*.

- The following animation shows the results of
*simulation*of 20*customer arrivals*and the*inter-arrival times*, starting with the prior**Γ(9,27)**.

- The left bottom plot visualizes the
*customers arrivals*.

- Every time a new datapoint is received (the next customers arrives at the shop), the
*prior belief*is updated.

- The right bottom table represents the summary statistics. Prior and Posterior means (of the arrival rate) respectively correspond to the previous and updated beliefs about the customers arrival rate at the shop.

- The next animation shows the same results starting with a
.*vague prior*Γ(ϵ,ϵ)

Advertisements