n - of Marcus Hutter

Marcus Hutter - 16 - Loss Bounds for UniGeneralization: Continuous ProbabilityIn statistical parameter estimation one often has a continuous hyBernoulli(θ) process with unknown θ ∈ [0, 1]).∫M := {µ θ : θ ∈ IR d }, ξ(x 1:n ) := dθ w(θ)·µ θ (x 1:n ),IR dThe only property of ξ needed was ξ(x 1:n )≥w µi ·µ i (x 1:n ) whichdropping the sum over µ i . Here, restrict the integral over IR d toθ. For sufficiently smooth µ θ and w(θ) we expectξ(x 1:n )> ∼ |N δn |·w(θ)·µ θ (x 1:n ) =⇒ D n< ∼ ln 1w µ+The average Fisher information ¯j n measures the curvature (paraln µ θ . Under some weak regularity conditions on ¯j n one can shoD n:= ∑ x 1:nµ(x 1:n ) ln µ(x 1:n)ξ(x 1:n )≤ ln 1w µ+ d 2 ln n 2π + 1 2 li.e. D n grows only logarithmically with n.