q,p ∥2 Hd = ⟨Ex∼q [s(x)k(x,⋅) + ∇x k(x,⋅)],Ex′∼q [s(x′)k(x′,⋅) + ∇x′ k(x′,⋅)]⟩ = Ex,x′∼q [⟨s(x)k(x,⋅) + ∇x k(x,⋅),s(x′)k(x′,⋅) + ∇x′ k(x′,⋅)⟩] = Ex,x′∼q [s(x)⊺⟨k(x,⋅),k(x′,⋅)⟩s(x′) + s(x)⊺⟨k(x,⋅),∇x′ k(x′,⋅)⟩ + ⟨∇x k(x,⋅),k(x′,⋅)⟩s(x′) + ⟨∇x k(x,⋅),∇x′ k(x′,⋅)⟩] = Ex,x′∼q [s(x)⊺k(x,x′)s(x′) + s(x)⊺∇x′ k(x,x′) + ∇x k(x,x′)⊺s(x′) + Tr(∇x,x′ k(x,x′))] ͕ಘΒΕΔɽҰํ ϕ(⋅) = ϕ∗ q,p (⋅) / ∥ϕ∗ q,p ∥Hd ΑΓ Ex∼q [Ap ϕ(x)] = Ex∼q [s(p)⊺ϕ∗ q,p (x) + ∇x ⋅ ϕ∗ q,p (x)] / ∥ϕ∗ q,p ∥Hd = Ex,x′∼q [s(x)⊺k(x,x′)s(x′) + s(x)⊺∇x′ k(x,x′) + ∇x k(x,x′)⊺s(x′) + Tr(∇x,x′ k(x,x′))] / ∥ϕ∗ q,p ∥Hd = ∥ϕ∗ q,p ∥Hd = D(q,p) ◻ 18