Fisher Information for Poisson Distribution

• 𝐼(𝜆) = 𝑛/𝜆

Derivation

the generic Fisher Information formula is as follows:

• 𝐼(𝜃) = - 𝐄[log-likelihood-function”(𝜃)]

we need the log-likelihood function 𝐿𝐿(𝜃) of a Poisson(𝜆) Distribution which is as follows

• 𝐿𝐿(𝜆) = 𝑙𝑛(𝜆)𝛴_{1≤𝑖≤𝑛}[𝑋_𝑖] - 𝑛𝜆 - 𝛴_{1≤𝑖≤𝑛} [𝑙𝑛(𝑋_𝑖!)] # click here for step-by-step computation 

now we need to compute the second derivative of 𝐿𝐿(𝜆)

• 𝐿𝐿’(𝜆) = (1/𝜆)𝛴_{1≤𝑖≤𝑛}[𝑋_𝑖] − 𝑛
• 𝐿𝐿”(𝜆) = -(1/𝜆²)𝛴_{1≤𝑖≤𝑛}[𝑋_𝑖]

plug into Fisher Information formula

• 𝐼(𝜃) = - 𝐄[log-likelihood-function”(𝜃)]
• 𝐼(𝜆) = - 𝐄[log-likelihood-function”(𝜆)] # the parameters (𝜃) of Poisson distribution is a single 𝜆
• 𝐼(𝜆) = - 𝐄[-(1/𝜆²)𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖)]
• 𝐼(𝜆) = (1/𝜆²) 𝐄[𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖)]
• 𝐼(𝜆) = (1/𝜆²) 𝐄[𝑋₁ + 𝑋₂+ … + 𝑋_𝑛]
• 𝐼(𝜆) = (1/𝜆²) ( 𝐄[𝑋₁] + 𝐄[𝑋₂] + … + 𝐄[𝑋_𝑛] )
• 𝐼(𝜆) = (1/𝜆²) [ 𝜆 + 𝜆 + … + 𝜆 ] # each 𝑋_𝑖 is a Poisson variable and its expected value is 𝜆
• 𝐼(𝜆) = 𝑛𝜆/𝜆²
• 𝐼(𝜆) = 𝑛/𝜆