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Distributions geometry

Consider the set of distributions P for a generic n̄-variate random vectorX, parameterized by l̄ parameters ξ ≡ (ξ1, . . . , ξl̄)

′, or X ∼ fξ. Then apoint p ∈ P reads

p ⇔ ξ ≡ (ξ1, . . . , ξl̄)′ ⇔ fξ (24.1)

A generic tangent vector h ∈ Rl̄ is defined as

h ≡ (h1, . . . , hl̄)′ ⇔ Yh(ξ) = h′∇ξ ln fξ(X) (24.2)

where ∇ξ is the gradient (39.17).

Example: univariate normal distribution

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Riemannian metric

A Riemannian metric g(ξ) is a family of l̄ × l̄ symmetric and positivedefinite matrices, one for each point ξ, such that the length of an arbitrarytangent vector h(ξ)

‖h(ξ)‖g ≡√h(ξ)′g(ξ)h(ξ) (24.6)

is independent of the coordinate system ξ.

Then the metric must be covariant: in terms of a different coordinatesystem ξ̃ ≡ (ξ̃1, . . . , ξ̃l̄)

′, i.e. ξ = ξ(ξ̃) for some invertible function ξ(·), itmust read as follows

gξ̃(ξ̃) = (∇ξ̃ξ)× gξ(ξ)× (∇ξ̃ξ)′ (24.7)

where [∇ξ̃ξ]l,s ≡ ∂ξ(ξ̃)/∂ξ̃l is the transpose Jacobian (39.23).

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Properties of the Riemannian metric

• The length of a smooth curve γ(s) for s ∈ [0, 1]

‖γ(·)‖g ≡∫ 1

0

√γ̇(s)′g(γ(s))γ̇(s)ds (24.8)

• The volume of a set S

volume(S) =

∫S

√det g(ξ)dξ1 · · · dξl̄ (24.9)

• The Fisher information matrix is a Riemannian metric

gFisher (ξ) ≡ Cv{∇ξ ln fξ(X)} (24.10)

Example: univariate normal distribution

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Properties of the Fisher informaton metric

• The Fisher metric is invariant if we transform the underlying randomvariables. Suppose that X̃ ≡ q(X) for some invertible function q, thenthe pdf becomes a new function

f̃ξ(x̃) =fξ(q

−1(x̃))√det(Jq(q−1(x̃)))2

(24.12)

Then∇ξ ln f̃ξ(X̃) = ∇ξ ln fξ(X) (24.13)

• The length (24.6) of a tangent vector (24.2) reads

‖h‖gFisher = Sd{h′∇ξ ln fξ(X)

}= Sd {Yh} (24.15)

Example: lognormal normal distributionExample: univariate normal distribution

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