Fisher Information

Fisher information

Assume there are two closely spaced stimuli s and s + $Δ$ s, then we can signal s if:

\frac{p (r | s + Δ s)}{p (r | s)} < 1

when the likelihood ratio (i.e. the left side) is equal to 1, it will be hard to decide if the stimulus is s or s + $Δ$ s.

If one want to be sure that he is signaling s, the likelihood ratio should be as different from 1 as possible (on average), which means the following value should be as large as possible:

< (\frac{p (r | s + Δ s)}{p (r | s)} - 1)^{2} >

after some derivations, we define the Fisher information based on it:

J (s) =< (\frac{d}{d s} l n p (r | s))^{2} >_{p (r | s)}