Uniqueness of the Fisher-Rao Metric on the Space of Smooth Densities

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14329
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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

Uniqueness of the Fisher-Rao Metric on the Space of Smooth Densities

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application/pdf Uniqueness of the Fisher-Rao Metric on the Space of Smooth Densities Martin Bauer, Martins Bruveris, Peter Michor

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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

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Uniqueness of the Fisher–Rao Metric on the Space of Smooth Densities Martin Bauer, Martins Bruveris and Peter W. Michor Brunel University London October 30, 2015 Theorem Let M be a compact manifold without boundary of dimension ≥ 2. Then any smooth weak Riemannian metric on the space Prob(M) of smooth positive probability densities, that is invariant under the action of the diffeomorphism group of M, is a multiple of the Fisher–Rao metric. Defintions Mm is a compact (orientable1) manifold without boundary. Diff(M) is the group of diffeomorphisms. The spaces of positve densities and probability densities are Dens+(M) = {µ ∈ Ωm (M) : µ(x) > 0 ∀x ∈ M} Prob(M) = µ ∈ Dens+(M) : M µ = 1 Their tangent spaces are Tµ Dens+(M) = Ωm (M) , Tµ Prob(M) = α ∈ Ωm (M) : M α = 0 The subgroup of Diff(M) preserving a density µ is Diff(M, µ) = {ϕ ∈ Diff(M) : ϕ∗ µ = µ} . 1 In this talk we assume M to be orientable. Main Theorem Let M be a compact manifold without boundary of dimension ≥ 2. Let G be a smooth bilinear form on Dens+(M) which is invariant under the action of Diff(M). Then Gµ(α, β) = C1(µ(M)) M α µ β µ µ + C2(µ(M)) M α · M β for some functions C1, C2 of the total volume µ(M). Corollary Let M be a compact manifold without boundary of dimension ≥ 2. Then any smooth weak Riemannian metric on the space Prob(M) of smooth positive probability densities, that is invariant under the action of the diffeomorphism group of M, is a multiple of the Fisher–Rao metric. Step 1: Moser trick Fix a basic probability density µ0. Any other µ can be written as c.µ0 = µ(M).µ0 = ϕ∗ µ , for some ϕ ∈ Diff(M) . By the Diff(M)-invariance of G, Gµ(α, β) = Gϕ∗µ(ϕ∗ α, ϕ∗ β) = Gc.µ0 (ϕ∗ α, ϕ∗ β) . It suffices to show that Gcµ0 (α, β) = C1(c). M α µ0 β µ0 µ0 + C2(c) M α · M β where Gcµ0 is invariant under Diff(M, µ0). Step 2: Distributions We intepret Gcµ0 as the bilinear mapping Gc : C∞ (M)×C∞ (M) → R , (f , g) → Gc(f , g) = Gcµ0 (f µ0, gµ0) , and consider the associated bounded mapping ˇGc : C∞ (M) → C∞ (M) = D (M) . Then it suffices to show that ˇGc(f ) = C1(c) f .µ0 + C2(c) M f µ0 µ0 . By Schwartz kernel theorem L (C∞(M), D (M)) ∼= D (M × M) via Gc(f , g) = ˇGc(f ), g = ˆGc, f ⊗ g . Step 3: Functions and Xexact(M, µ0) Denote the exact divergence free vector fields Xexact(M, µ0) = ˆι−1 µ0 (dω) : ω ∈ Ωm−2 (M) Lemma: If f ∈ C∞(M), U ⊆ M connected and (LX f )|U = 0 for all X ∈ Xexact(M, µ0), then f |U is constant. Proof: Let x ∈ U. In a local chart (Ux , u) µ0|Ux = du1 ∧ · · · ∧ dum and for g ∈ C∞ c (Ux ) with g = 1 near x, X = ˆι−1 µ0 (g.u2 .du3 ∧ · · · ∧ dum ) ∈ Xexact(M, µ0) and X = ∂u1 near x. Step 4: Distributions and Xexact(M, µ0) Lemma: If A ∈ D (M), U ⊆ M connected and LX A|U = 0 for all X ∈ Xexact(M, µ0), then A|U = Cµ0|U for some constant C, i.e. A, f = C M f µ0 , ∀f ∈ C∞ c (U) . Proof: It is enough to show A, g = 0, if M gµ0 = 0. We can assume U is diffeomorphic to Rm. Then M gµ0 = 0 ⇒ gµ0 = dα for some α ∈ Ωm−1 c (U) . Write α = j fj dβj and Xj = ˆι−1 µ0 (dβj ) ∈ Xexact(M, µ0) . Then g = j LXj fj and so A, g = j A, LXj fj = − j LXj A, fj = 0 . Step 5: Locally constant functions Invariance of ˇGc is LX ˇGc(f ) = ˇGc(LX f ) for X ∈ X(M, µ0). Lemma: If f |U is constant and U ⊆ M is open and connected, then ˇGc(f )|U = CU(f )µ0|U , for some constant CU(f ). Proof. Let X ∈ Xexact(M, µ0), x ∈ U. Construct Y ∈ Xexact(M, µ0) with Y = X on M \ U and Y = 0 near x . Then LX ˇGc(f ) = ˇGc(LX f ) = ˇGc(LY f ) = LY ˇGc(f ) . Y vanishes near x and so does LX ˇGc(f ). Hence LX ˇGc(f )|U = 0. Step 6: Off-diagonal support Lemma: The distribution ˆGc − C2µ0 ⊗ µ0 ∈ D (M × M) is supported on the diagonal of M × M for some C2. Proof. If f and g are functions with disjoint support, then Gc(f , g) = ˇGc(f ), g = C(f ) M gµ0 = ˇGT c (g), f = C(g) M f µ0 . From this we can conclude that ˆGc, f ⊗ g = C2 µ0 ⊗ µ0, f ⊗ g , Gc(f , g) = C2 M f µ0 M gµ0 , or equivalently ˆGc − C2µ0 ⊗ µ0 is supported on the diagonal. Step 7: Putting it together Assume that ˆGc ∈ D (M × M) is supported on the diagonal. Then ˇGc(f ) = |α|≤k Aα.∂α f or Gc(f , g) = |α|≤k Aα, (∂α f ) .g holds in local charts (U, u) with Aα ∈ D (U) and the operators Aα are uniquely determined. Choose X ∈ Xexact(M, µ0) with X|U = ∂ui . By invariance 0 = Gc(LX f , g) + Gc(f , LX g) = α Aα, (∂α ∂ui f ).g + (∂α f )(∂ui g) = α Aα, ∂ui ((∂α f ).g) = α −∂ui Aα, (∂α f ).g . From this we conclude that ∂ui Aα|U = 0 for each α, and each i. From ∂ui Aα|U = 0 we conclude that Aα|U = Cαµ0|U and so Gc(f , g) = U (Lf ).gµ0 , L = |α|≤k Cα∂α . Set g = 1. By invariance for X ∈ X(M, µ0), 0 = Gc(LX f , g) + Gc(f , LX g) = U L(LX f ).1.µ0 + 0 , and so the distribution h → U L(h)µ0 vanishes on functions of the form LX f . From this we can conclude that U L(f )µ0 or L = C Id. Hence ˆGc(f ⊗ g) = C M fgµ0 . This concludes the proof. Thank you for your attention.