Solution
Information in a Single Bit
Show the problem again
X is Poisson with unknown rate λ. Instead of X itself you are told only the bit Y = 1 if X = 0 and Y = 0 otherwise. Compute the Fisher information about λ contained in Y when λ = ln 3. Give a reduced fraction.
Worked solution
Y is Bernoulli with success probability q(λ) = e^(−λ), so q = 1/3 at λ = ln 3. For a Bernoulli parameter q(λ), the Fisher information is (q′)²/(q(1 − q)). Here q′ = −e^(−λ) = −q, so the information is q²/(q(1 − q)) = q/(1 − q) = (1/3)/(2/3) = 1/2. For comparison the full observation X carries 1/λ = 1/ln 3 ≈ 0.91, so the single bit retains a bit over half of it. As a check, the score of Y is −1 when Y = 1 (probability 1/3) and q/(1 − q) = 1/2 when Y = 0; its second moment is 1/3 + (2/3)(1/4) = 1/2. The answer is 1/2.
Source: Original problem written for AxiomIQ; Fisher information under data reduction.