Datasets:
ROmondi
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STST

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i,j=1
d
i,j=1
wij (EF [|Xi −Xj|p] −|Yi −Yj|p)2
wij EF [|Xi −Xj|p]2 −2EF [|Xi −Xj|p]|Yi −Yj|p +|Yi −Yj|2p
wij EF [|Xi −Xj|p]2 −2EF [|Xi −Xj|p]EG[|Yi −Yj|p]+EG[|Yi −Yj|2p] .
i,j=1
B.5 Logarithmic score
For any F,G ∈ P(Rd) such that F and G have probability density functions that belong to L1(Rd), the
expectation of the logarithmic score is analogous to its univariate version :
EG[LogS(F,Y )] = DKL(G||F)+H(F),
where DKL(G||F) is the Kullback-Leibler divergence from F to G and H(F) is the Shannon entropy of F.
DKL(G||F) =
H(F) =
B.6 Hyvärinen score
g(y)log g(y)
Rd
Rd
f(y) dy
f(y)log(f(y))dy.
For F,G ∈ P(Rd) such that their probability density functions f and g such that they are twice continuously
differentiable and satisfying ∇f(x) → 0 and ∇g(x) → 0 as ∥x∥ → ∞, the expectation of the Hyvärinen score
is :
E[HS(F,Y )] =
g(y)⟨∇log(f(y)) − 2∇log(g(y)),∇log(f(y))⟩g(y)dy
Rd
where ∇ is the gradient operator and ⟨·,·⟩ is the scalar product. The proof is similar to the proof for the
univariate case using integration by parts and Stoke’s theorem (Parry et al., 2012).
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B.7 Quadratic score
For any F,G ∈ L2(Rd), the expectation of the quadratic score is analogous to its univariate version :
EG[QuadS(F,Y )] = ∥f∥2
2 −2⟨f,g⟩,
where ⟨f,g⟩ = Rd
f(y)g(y)dy.
B.8 Pseudospherical score
For any F,G ∈ Lα(Rd), the expectation of the quadratic score is analogous to its univariate version :
EG[PseudoS(F,Y )] = −⟨fα−1,g⟩
where ⟨fα−1,g⟩ = Rd
f(y)α−1g(y)dy.
C Proofs
C.1 Proposition 1
∥f∥α−1
α
,
Proof of Proposition 1. Let F ⊂ P(Rd) be a class of Borel probability measure on Rd and let F ∈ F be a
forecast and y ∈ Rd an observation. Let T : Rd → Rk be a transformation and let S be a scoring rule on Rk
that is proper relative to T(F) = {L(T(X)),X ∼ F ∈ F}.
EG[ST(F,Y )] = EG[S(T(F)),T(Y ))]
=ET(G)[S(T(F),Y )]
Given that T(F),T(G) ∈ T(F) and S is proper relative to T(F),
ET(G) [S(T(G),Y )] ≤ ET(G)[S(T(F),Y )]
⇔ EG[ST(G,Y)] ≤ EG[ST(F,Y)]
(23)
Proof of the strict propriety case in Proposition 1. The notations are the same as the proof above except the
following. Let T : Rd → Rk be an injective transformation and let S be a scoring rule on Rk that is strictly
proper relative to T(F) = {L(T(X)),X ∼ F ∈ F}.
The equality in Equation (23) leads to :
EG[ST(G,Y )] = EG[ST(F,Y )]
⇔ EG[S(T(G),T(Y ))] = EG[S(T(F),T(Y ))]
⇔ ET(G)[S(T(G),Y )] = ET(G)[S(T(F),Y )]
The fact that S is strictly proper relative to T(F) leads to T(F) = T(G), and finally since T is injective, we
have F = G.
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C.2 Proposition 3
Proof of Proposition 3. The proof relies on the reproducing kernel Hilbert space (RKHS) representation of
the kernel scoring rule Sρ. For a background on kernel scoring rule, maximum mean discrepancies and RKHS,
we refer to Smola et al. (2007) or Steinwart and Christmann (2008, Section 4).
Let Hρ denote the RKHS associated with ρ. We recall that Hρ contains all the functions ρ(x,·) and that
the inner product on Hρ satisfies the property
⟨ρ(x1,·),ρ(x2,·)⟩Hρ
= ρ(x1,x2).
The kernel mean embedding is a linear application Ψρ : Pρ → Hρ mapping an admissible distribution F ∈ Pρ
into a function Ψρ(F) in the RKHS and such that the image of the point measure δx is ρ(x,·). Equation (16)
giving the kernel scoring rule for an ensemble prediction F = 1
M
M
m=1δxm
can be written as
Sρ(F,y) = 1
2 ⟨Ψρ(F) −Ψρ(δy),Ψρ(F)−Ψρ(δy)⟩Hρ
= 1
2∥Ψρ(F −δy)∥2
Hρ
.
The properties of the kernel mean embedding ensure that this relation still holds for all F ∈ Pρ. As a
consequence, if (Tl)l≥1 is an Hilbertian basis of Hρ, we have
Sρ(F,y) = 1
2∥Ψρ(F −δy)∥2
Hρ
= 1
2 l≥1
⟨Ψρ(F −δy),Tl⟩2
Hρ
.
Finally, the properties of the kernel mean embedding ensure that, for all T ∈ Hρ,
⟨Ψρ(F −δy),T⟩Hρ