I think the term “forecaster” is perhaps confusing here and it would be more clear to say “what fraction of the time is the final configuration in terms of left/right the same under a tiny random perturbation”.
That is, let θ be the fixed initial random configuration and let is_left(θ) be whether after 20 seconds the final configuration (when starting from θ) is to the left. (Note that this is a fully deterministic quantity.) Then, let m(θ,X) pertub a particle in the state θ as described given the random variable X.
Now, we care about the value of:
EX[is_left(m(θ,X))]
(In particular whether it is very close to 0.5 or is >0.51 or <0.49.)
I think the term “forecaster” is perhaps confusing here and it would be more clear to say “what fraction of the time is the final configuration in terms of left/right the same under a tiny random perturbation”.
That is, let θ be the fixed initial random configuration and let is_left(θ) be whether after 20 seconds the final configuration (when starting from θ) is to the left. (Note that this is a fully deterministic quantity.) Then, let m(θ,X) pertub a particle in the state θ as described given the random variable X.
Now, we care about the value of:
EX[is_left(m(θ,X))]
(In particular whether it is very close to 0.5 or is >0.51 or <0.49.)
agree, I changed the question wording