STAT314. Poisson MLE. Derivation.
STAT314. Poisson MLE. Derivation.
Elena Moltchanova
2021S2
Let xi|µ ∼ Pois(µ) for i = 1, …, n. Then the joint likelihood (i.e., the product of individual likelihoods):
L = f(x|µ) = f(x1, …, xn|µ) =
∏
i
f(xi|µ) =
∏
i
µxie−µ
xi!
=
µ
∑
i
xie−nµ∏
i xi!
.
for µ > 0.
First, let’s log the likelihood to make things easier:
logL =
∑
i
xi logµ− nµ− log
(∏
i
xi!
)
.
Now, let’s differentiate with respect (w.r.t.) to µ:
∂ logL
∂µ
=
∑
i
xi
1
µ
− n.
Set it to 0, and solve for µ:
∑
i
xi
1
µ
− n = 0 ⇐⇒ µ̂ =
∑
i xi
n
= x̄.
Remember, that that is not yet a point where the function reaches maximum. It may be a minimum or an
inflection point. Always check for the second order condition:
∂2 logL
∂µ2
=
∑
i
xi
(
−
1
µ2
)
.
You can substitute specific µ̂ to find that
∑
i
xi
(
−
1
µ̂2
)
= −
∑
i xi
x̄2
= −
nx̄
x̄2
= −
n
x̄
< 0 ∀x̄ > 0.
Or you can use the fact that µ2 > 0 for all µ > 0 to arrive at the same conclusion. Thus m̂u = x̄ is the
maximum likelihood estimate (MLE) of the Poisson intensity parameter µ.
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