Home > Single parameter > Poisson mean example, part II

## Poisson mean example, part II

In the last post, I described the process of determining a prior for $\lambda$, my son’s average number of text messages per day.   I decided to model my beliefs with a gamma(a, b) density with a = 44.4 and b = 4.4.

Now I observe some data.  I looked at the online record of text messaging for the first seven days that’s been at school and observe the counts

Sat  Sun  Mon  Tue  Wed  Thu  Fri
19     4      26      17    15       0    17

If we assume these counts are Poisson with mean $\lambda$, then the likelihood function is given by

L(lambda) =  lambda^s  exp(-n lambda),

where $n$ is the number of observations (7) and $s$ is the sum of the observations ( 98).

The posterior density (using the prior times likelihood recipe) is given by

L(lambda) x g(lambda) = lambda^{a+s-1}  exp(-(b+n) lambda),

which we recognize as a gamma density with shape $a_1 = a + s = 142.4$ and rate $b_1 = b + n = 11.4$.

Using the following R commands, we construct a triplot (prior, likelihood, and posterior).

> a=44.4; b=4.4
> s=sum(y); n=length(y)

> a=44.4; b=4.4
> s=sum(y); n=length(y)
> curve(dgamma(x,shape=s+a,rate=n+b),col=”red”,xlab=”LAMBDA”,
+   ylab=”DENSITY”,lwd=3,from=3,to=25)
> curve(dgamma(x,shape=a,rate=b),col=”blue”,lwd=3,add=TRUE)
> curve(dgamma(x,shape=s+1,rate=n),col=”green”,lwd=3,add=TRUE)
> legend(“topright”,c(“PRIOR”,”LIKELIHOOD”,”POSTERIOR”),
+  col=c(“blue”,”green”,”red”),lty=1,lwd=3)

Note that (1) the posterior is a compromise between the data information (likelihood) and the prior  and (2) the posterior is more precise since we are combining two sources of information.
I would like to predict my son’s text message use in the next month.  Let $y_1^*, ..., y_{30}^*$ denote the number of text messages in the next month.  The total number of messages $s^*$ has, conditional on $\lambda$, a Poisson distribution with mean $30 \lambda$.
I am interested in computing the posterior predictive  distribution for $s^*$.  One can simulate this distribution by (1) simulating a value of the parameter $\lambda$ from the gamma(142.4, 11.4) posterior and then (2) simulating $s^*$ from a Poisson distribution with mean $30 \lambda$.
We can do this simulation in R in two lines.   In the last line, I tabulate the values of $s^*$ and plot the counts.
> lambda=rgamma(1000,shape=142.4,rate=11.4)
> ys=rpois(1000,30*lambda)
> plot(table(ys),ylab=”FREQUENCY”)
Looking at the graph, it is likely that my son will have between 310 and 450 text messages in the next 30 days.
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