# poisBayes: Bayesian Prediction Limits for Poisson Distribution (Gamma... In plpoisson: Prediction Limits for Poisson Distribution

## Description

The function provides the Bayesian prediction limits of a Poisson random variable derived based on a gamma prior. The resulting prediction bounds quantify the uncertainty associated with the predicted future number of occurences in a time window of size t.

## Usage

 `1` ```poisBayes(xobs, n, s, t, a, b, alpha = 0.05) ```

## Arguments

 `xobs` a numeric value denoting the number of the observed occurrencies. `n` a numeric value representing the total number of the time windows `s` in the past (observed time windows). `s` a numeric value corresponding to the fixed size (or average size) of the observed time windows. `t` a numeric value indicating the size of the future time window. `a` a poisitive real number denoting the shape hyperparameter of a gamma prior distribution. `b` a poisitive real number representing the rate hyperparameter of a gamma prior distribution. `alpha` a numeric value associated to the credible probability. By default `alpha = 0.05`, thus an prediction interval at 95% will be returned.

## Details

When the argument `b = Inf`, one can obtain prediction limits with uniform prior by setting the argument `a = 1`. Similarly, one can get the limits with a Jeffreys prior by setting the argument `a = 0`.

## Value

A list containing the following components:

 `lower` An integer value representing the lower bound of the prediction limit. `upper` An integer value representing the upper bound of the prediction limit.

## Author(s)

Valbona Bejleri, Luca Sartore and Balgobin Nandram

## References

Bejleri, V., & Nandram, B. (2018). Bayesian and frequentist prediction limits for the Poisson distribution. Communications in Statistics-Theory and Methods, 47(17), 4254-4271.

Bejleri, V. (2005). Bayesian Prediction Intervals for the Poisson Model, Noninformative Priors, Ph.D. Dissertation, American University, Washington, DC.

`poiss`, `poisJEFF`, `poisUNIF`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```# Loading the package library(plpoisson) set.seed(2020L) # Number of observed time windows n <- 555L # Simulating a dataset data <- cbind.data.frame( occ_obs = rpois(n, rgamma(n, 5.5, .5)), win_siz = rgamma(n, 1.44, .777) ) ## Bayesian prediction limits ## (with gamma prior) poisBayes(sum(data\$occ_obs), # Past occurrencies nrow(data), # Total past time windows mean(data\$win_siz), # Window size 333, # Size of future window 2, 2.22) # Hyper-parameters for gamma prior ```