3.4 The Poisson distribution

3.4.1 Conjugate prior

A discrete random variable x is said to have a Poisson distribution of mean λ if it has the density

This distribution often occurs as a limiting case of the binomial distribution as the index and the parameter but their product (see Exercise 6 in Chapter 1). It is thus a useful model for rare events, such as the number of radioactive decays in a fixed time interval, when we can split the interval into an arbitrarily large number of sub-intervals in any of which a particle might decay, although the probability of a decay in any particular sub-interval is small (though constant).

Suppose that you have n observations from such a distribution, so that the likelihood is

where T is the sufficient statistic

We have already seen in Section 2.10 on ‘Conjugate Prior Distributions’ that the appropriate conjugate density is

that is, , so that λ is a multiple of a chi-squared random variable. Then the posterior density is

that is

where

3.4.2 Reference prior

This is a case where we can try to use Jeffreys’ rule. The log-likelihood resulting from a single observation x is

so that

and hence,

Consequently Jeffreys’ rule suggests the prior

which corresponds to , S0=0 in the conjugate family, and is easily seen to be equivalent to a prior uniform in . It may be noted that there is a sense in which this is intermediate between a prior uniform in and one uniform in λ itself, since as

so that there is a sense in which the transformation from λ to can be regarded as a ‘zeroth power’ transformation (cf. Box and Cox, 1964).

On the other hand, it could be argued that λ is a scale parameter between 0 and and that the right reference prior should therefore be

which is uniform in and corresponds to , S0=0 in the conjugate family. However, the difference this would make in practice would almost always be negligible.

3.4.3 Example

The numbers of misprints spotted on the first few pages of an early draft of this book were

It seems reasonable that these numbers should constitute a sample from a Poisson distribution of unknown mean λ. If you had no knowledge of my skill as a typist, you might adopt the reference prior uniform in for which , S0=0. Since

your posterior for λ would then be , that is, . This distribution has mean and variance

Of course, I have some experience of my own skill as a typist, so if I considered these figures, I would have used a prior with a mean of about 3 and variance about 4. (As a matter of fact, subsequent re-readings have caused me to adjust my prior beliefs about λ in an upwards direction!)  If then I seek a prior in the conjugate family, I need

which implies and S0=1.5. This means that my posterior has , S1=13.5 and so has mean and variance

The difference between the two posteriors is not great and of course would become less and less as more data were included in the analysis. It would be easy enough to give HDRs. According to arguments presented in other cases, it would be appropriate to use HDRs for the chi (rather than the chi-squared distribution), but it really would not make much difference if the regions were based on HDRs for chi-squared or on values of chi-squared corresponding to HDRs for log chi-squared.

3.4.4 Predictive distribution

Once we know that λ has a posterior distribution

then since

it follows that the predictive distribution

(dropping a factor which depends on alone). Setting , you can find the constant by reference to Appendix A. In fact, at least when is an integer, the predictive distribution is negative binomial, that is

Further, although this point is not very important, it is not difficult to see that the negative binomial distribution can be generalized to the case where is not an integer. All we need to do is to replace some factorials by corresponding gamma functions and note that (using the functional equation for the gamma function)

so that you can write the general binomial coefficient as

The negative binomial distribution is usually defined in terms of a sequence of independent trials each of which results in success or failure with the same probabilities π and (such trials are often called Bernoulli trials) and considering the number x of failures before the nth success. We will not have much more use for this distribution in this book, but it is interesting to see it turning up here in a rather different context.