## Poisson Distribution (2)

 LA home Computing MML  Discrete   Integers    Poisson Also see  Geometric  Poisson1
P(x|α) = e αx / x!,    integer x≥0,
has one parameter, α, and mean = variance = α.
Note, P(x|α) = P(x-1|α) . α / x
so P(x|α) increases with x while x<α and decreases when x>α.

Given n data, x1, x2, ..., xn, the likelihood
P(x1, x2, ..., xn | α) = e-n.α∑xi / (x1! ... xn!)

neg log likelihood,
L = - log P(x1, x2, ..., xn | α)
= n.α - (∑xi)logα + log x1! + ... + log xn!
1st derivative
d L / d α = n - (∑xi)/α
 Equating this to zero, αmaxLH = (∑xi) / n.
2nd derivative
d2 L / d α2 = (∑xi) / α2
which has expectation, i.e., Fisher information, Fα = nα/α2 = n/α,
note   +log Fα = log n - log α

Assume prior,   h α = (1/A).e-α/A,   which has mean A.
Note   - log(h α) = log A + α/A.

Message length,
m = - log(h α) + L + 1/2 log Fα + (-log 12 +1)/2

To estimate α, differentiate m with respect to α
d m / d α = 1/A + n - (∑xi)/α - 1/(2α)
equate to zero
αMML = (∑xi + 1/2) / (n + 1/A)
uncertainty region sqrt(12/FαMML) = sqrt(12 αMML / n)
-- LA, 3/7/2007
Some sanity checks:
1. If it happens that x1 = ... = xn = x then αMML -> x, and the uncertainty region ->0 as n->∞.
2. If x1 = ... = xn = 0 then αMML->0 as n->∞.
3. αMML -> αmaxLH = ∑xi/n as n->∞.
 A=, data=1 2 3 4 5 6 7 8 9 10 11 12 13 , op=needs JavaScript turned on (Also see Geometric.)

See [IP 1.2] for an implementation.