Poisson Distribution (1)

The Poisson distribution with parameter α>0, for n≥0:

P(n) =

e-α αn n!

n≥0

P(n) = (α/n) P(n-1), so P(n) increases while n<α and decreases when n>α, i.e. the mode is α; this is also the maximum likelihood estimate of α given observed data `n'.

The Poisson distribution can be derived (e.g. Meyer 1970) as the distribution of the number of particle decays in a radioactive source in unit time where α is the rate, i.e. where the probability of a decay in a small time interval, dt, is α.dt.

MML

We observe a value of `n' (e.g. n decays in unit time). The negative log likelhood, i.e. -log P(n) is

-log(P(n|α))

= α - n.log α + log n!

The second derivative with respect to the parameter α is n/α².
The expectation of this over n, i.e. the Fisher information, is

α/α² = 1/α

-log(h(α)) -log(p(n|α)) + 1/2 log F(α) +(-log 12 + 1)/2

d/d α {	-log 1/A + α/A		//from h
	+ α - n.log α + log n!		//from likelihood
	- 1/2 log α		//from F
	+ (-log 12 +1)/2 }

= 1/A + 1 - n/α - 1/(2.α)

= 1/A + 1 - (n+1/2)/α

= 0

Poisson Process

The Poisson process models, for example, the number of radioactive decays in a given time t:

P(n,t) =

e-α.t(α.t)n n!

n≥0

Easier

For a more convenient to use MML formulation, with a JavaScript implementation, and for application to datasets of several data values, see [here(click)].

Notes

• [WD97] C. S. Wallace & D. L. Dowe. MML Mixture Modelling of Multi-state, Poisson, von Mises Circular and Gaussian Distributions. Proc. 6th Int. Workshop on Artificial Intelligence and Statistics, pp.529-536, 1997.