1Vθ（x）π（x）,.PDF

Supplementary Material for Adaptive Metropolis with Online Relabeling Notations and assumptions By convention, vectors x ∈ Rd are column vectors. xT is the transpose vector of x. We fix a norm · on vectors and will also denote by · the derived norm for matrices. Let π be a probability density with respect to (w.r.t.) the Lebesgue measure on X where X ⊆ Rd is measurable (Rd is endowed with its Borel σ-field). It is assumed that (a) π in invariant under the action of P , a finite group of d × d block permutation matrices. (b) π has finite second moment. Let C + be the set of the real d × d (symmetric) positive definite matrices. For any θ = (µ, Σ) ∈ Rd × C+ and x ∈ X , d d define the quadratic loss Lθ (x) = (x − µ)T Σ−1 (x − µ) (1) and the set Vθ = {x ∈ X : Lθ (x) = min Lθ (P x)}. P ∈P Observe that for any θ ∈ Rd × C+, Vθ is measurable. d For any θ ∈ Rd × C+, let πθ be the probability density on Rd , defined by d πθ (x) = Z −1 1Vθ (x)π (x), where Zθ = π (x)dx. θ Vθ Under the assumptions on π, πθ has an expectation µπθ = x πθ (x)dx and a covariance matrix Σπθ = (x − µπθ )(x − µπθ )T πθ (x)dx. Define the function w : Rd × C+ → R by d w (θ) = − log N (x |θ) πθ (x)dx. Finally, d