Home > Priors, Single parameter > Constructing a prior for a Poisson mean

Constructing a prior for a Poisson mean

I am interested in learning about my son’s cell phone use.   Suppose the mean number of text messages that he sends and receives per day is equal to \lambda.   I will observe the number of text messages y_1, ..., y_n for n days — we’ll assume that conditional on \lambda, y_1, ...., y_n are independent Poisson(\lambda).

How do I construct a prior density on \lambda?   Suppose I model my beliefs by use of a gamma density with shape parameter a and rate parameter b.  I want to figure out the values of the prior parameters a and b.

There are two distinct ways of assessing your prior.  One can think about plausible values of the mean number of daily text messages \lambda.  Alternatively, it may be easier to think about the actual number of text messages y — if we assume a gamma prior, then one can show that the predictive density of y is given by

f(y) = \frac{\Gamma(a+y)}{\Gamma(a) y!} \frac{b^a}{(b+1)^{a+y}}, y= 0, 1, 2, ...

Personally, I think it is easier to think about the number of text messages y.  My best prediction at y is 10.  After some additional thought, I decide that my standard deviation for y is equal to 3.5.    One can show that the mean and standard deviation of the predictive density are given by

E(y) = \frac{a}{b}, SD(y) = \sqrt{\frac{a}{b} + \frac{a}{b^2}}

If one matches my guesses with these expressions, one can show a = 44.4, b = 4.4.

To see if these values make any sense, I plot my predictive density.  This density is a special case of a negative binomial density where (using the R notation)

size = a,  prob = b /(b + 1).

I graph using the following R commands.  One can compute P(5 <= y <= 15) = 0.89 which means that on 89% of the days, I believe Steven will send between 5 and 15 messages.

a = 44.4; b = 4.4
plot(0:30,dnbinom(0:30,size=a,prob=b/(b+1)),type=”h”,
xlab=”y”, ylab=”Predictive Prob”, col=”red”, lwd=2)

a = 44.4; b = 4.4
plot(0:30,dnbinom(0:30,size=a,prob=b/(b+1)),type=”h”,
xlab=”y”, ylab=”Predictive Prob”, col=”red”, lwd=2)

poisson

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Categories: Priors, Single parameter
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