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Some remarks on the intrinsic Cramer-Rao bound

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Some remarks on the intrinsic Cramer-Rao bound GSI 2013, Paris Axel Barrau* and Silv`ere Bonnabel Mines Paristech august, 29th (Mines Paristech) august, 29th 1 / 21 Introduction Problem: estimate a covariance matrix Σ given a sample of X ∼ N(0, Σ) Sample covariance matrix estimation. S.T. Smith has proven 1 for the natural distance d on the cone of covariance matrices: E(d2 (Σ, ˆΣ)) Σ with Σ = Cste. This result can be seen as: A consequence of information geometry A consequence of the invariances of the problem 1 S. T. Smith, ”Covariance, Subspace, and Intrinsic Cramer-Rao Bounds” IEEE Trans. signal process., vol. 53, no. 5, May 2005 (Mines Paristech) august, 29th 2 / 21 1 Cramer-Rao bound in classical estimation theory 2 Intrinsic Cramer-Rao bound 3 Invariant parametric families (Mines Paristech) august, 29th 3 / 21 Cramer-Rao bound in classical estimation theory Cramer-Rao bound in classical estimation theory (Mines Paristech) august, 29th 4 / 21 Cramer-Rao bound in classical estimation theory Cramer-Rao bound Parametric family of densities: p(x|θ), θ ∈ Rn Classical Fisher Information Matrix: Ii,j (θ) = E( ∂ ∂θi log p(x|θ) ∂ ∂θj log p(x|θ)) Cramer-Rao Lower Bound for unbiased estimators: Var(ˆθ) I−1(θ) (Mines Paristech) august, 29th 5 / 21 Cramer-Rao bound in classical estimation theory Fisher Metric Definition The Fisher metric is the Riemannian metric defined by the local scalar product dθT I(θ)dθ. Definition The Fisher distance is the geodesic distance associated to the Fisher metric. (Mines Paristech) august, 29th 6 / 21 Cramer-Rao bound in classical estimation theory Examples: Location parameter: p(x|θ) = f (x − θ) The Fisher distance is proportional to the euclidian distance. Scale parameter: p(x|θ) = 1 θ f ( x θ ) The Fisher distance is proportional to d(θ1, θ2) = || log(θ1)−log(θ2)||. (Mines Paristech) august, 29th 7 / 21 Intrinsic Cramer-Rao bound Intrinsic Cramer-Rao bound (Mines Paristech) august, 29th 8 / 21 Intrinsic Cramer-Rao bound Normal coordinates θ ∈ M endowed with a Riemannian metric gθ. An orthogonal basis X1, ...Xn of the tangent plane defines a set of local coordinates through (a1, ..., an) → expθ(a1X1 + ... + anXn). gθ becomes the euclidian scalar prduct. (Mines Paristech) august, 29th 9 / 21 Intrinsic Cramer-Rao bound Basic statistical tools 2 The exponential coordinates map M to its tangent plane at θ. Bias of an estimator ˆθ: b(θ) = E(exp−1 θ (ˆθ)) Covariance of an estimator ˆθ: C(θ) = Cov(exp−1 θ (ˆθ)) 2 X. Pennec, ”Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements” Journal of Mathematical Imaging and Vision, 25:127-164, 2006 (Mines Paristech) august, 29th 10 / 21 Intrinsic Cramer-Rao bound Examples: Estimation of a covariance matrix in statistics Subspace estimation in signal processing Pose estimation in robotics (Mines Paristech) august, 29th 11 / 21 Intrinsic Cramer-Rao bound Intrinsic Cramer-Rao bound The Intrinsic Fisher Information Matrix is defined using local coordinates: Ii,j (θ) = E( ∂ ∂θi log p(x|θ) ∂ ∂θj log p(x|θ)) Intrinsic Cramer-Rao lower bound without bias: C(θ) Ii,j (θ)−1 + curvature terms (Mines Paristech) august, 29th 12 / 21 Intrinsic Cramer-Rao bound Intrinsic root mean square error Let d(., .) denote the riemannian distance on M. Definition 2 θ = E(d(θ, ˆθ)2 )) (= E(|| exp−1 θ (ˆθ)||2 ) = E(Tr(exp−1 θ (ˆθ) exp−1 θ (ˆθ)T ) = Tr[C(θ)]) If d is the Fisher distance: I(θ) = Id Neglecting the curvature terms: C(θ) I(θ)−1 = Id 2 θ = Tr(C(θ)) n (Mines Paristech) august, 29th 13 / 21 Intrinsic Cramer-Rao bound Application Sample Covariance Matrix estimation: p(x|Σ) = N(0, Σ) The Fisher metric is the natural metric: GΣ(D, D) = Tr(DΣ−1 )2 As proved by Smith: 2 n(n + 1) 2 which doesn’t depend on Σ. (Mines Paristech) august, 29th 14 / 21 Invariant parametric families Invariant parametric families (Mines Paristech) august, 29th 15 / 21 Invariant parametric families Invariances Consider a parametric family p(x|θ). Assume there exist two actions of a group G: (g, x) → φg (x) is an action of G on X. (g, θ) → ρg (θ) is an action of G on M. Definition Invariance under the action of G : y = φg (x) has density function py (y|ρg (θ)) = px (x|θ) (Mines Paristech) august, 29th 16 / 21 Invariant parametric families Example: radioactive decay. Law: p(t|θ) = 1 θ exp(− t θ ) This law has to be insensitive to a change of units (for instance from minuts to seconds): θ → Θ = 60 × θ t → T = 60 × t pT (T|Θ) = px (T|Θ) (Mines Paristech) august, 29th 17 / 21 Invariant parametric families Properties of invariant families Proposition If p(x|θ) is invariant under the actions ρg and φg of a group G and if ρg is transitive, then ∀(θ, g), ρg (θ) = θ Corollary If p(x|θ) is invariant under the actions ρg and φg of a group G and if ρg is transitive, then the Cramer-Rao Bound on the Mean Square Error associated to any G-invariant metric on M is constant. (Mines Paristech) august, 29th 18 / 21 Invariant parametric families Examples: Wahba’s problem : Etimate R using noisy measurments Yi = RT bi + Wi For any right-invariant metric we have: 2 R = Cste Sample Covariance Matrix estimation: p(x|Σ) = 1 (2π) n 2 exp(− 1 2 xT Σ−1 x) The family is invariant under the action of GLn(R) ρA(Σ) = AΣAT . As the natural metric of the cone of covariance matrices has the same invariance we have: 2 Σ = Cste (Mines Paristech) august, 29th 19 / 21 Invariant parametric families Conclusions The constant lower bound found by Smith has two interpretations: It is a general property of the Fisher metric. It is a consequence of the invariances of the problem. Furether result: An optimal estimator respects the invariances of the system. (Mines Paristech) august, 29th 20 / 21 Invariant parametric families Questions ? (Mines Paristech) august, 29th 21 / 21