Page - 315 - in Differential Geometrical Theory of Statistics
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Entropy2017,19, 7
3. EscortDistributionsandGeneralizationsofExpectations
Inanomalousstatistics, ageneralizedexpectation, calledanescortexpectation, isoftendiscussed
since the standard expectation does not exist in general (cf. [2,5,6]). In this section, we recall
generalizationsofexpectationsandintroduceasequential structureofescortdistributions.
LetSqbeaq-exponential family. Foragiven p(x;θ)∈Sqwedefinetheq-escortdistributionPq(x;θ)
of p(x;θ)andthenormalizedq-escortdistributionPescq (x;θ)by
Pq(x;θ) := Pq,(1)(x;θ) := {p(x;θ)}q,
Pescq (x;θ) := 1
Zq(p) {p(x;θ)}q, where Zq(p)= ∫
Ω {p(x;θ)}qdx,
respectively. For a q-exponential familySq = {pq(x;θ)}, the set ofnormalizedescortdistributions
Sq′={Pescq (x;θ)} isaq′-exponential familywithq′=(2q−1)/q.
Example 2. Let tν(x;μ,σ) be a univariate Student’s t-distribution with degree of freedom ν. Then its
normalized escort distribution is also a univariate Student’s t-distribution with degree of freedom ν+2.
In fact, fromEquation (4), adirect calculationshows that
q′= 2q−1
q = ν+5
ν+3 .
This implies that thedegree of freedomν′= ν+2. Therefore,weobtaina sequenceof escortdistributions froma
givenStudent’s t-distribution tν:
tν→ tν+2→ tν+4→··· .
This sequence is called a τ-sequence, and the procedure to obtain from a given t-distribution to another
t-distribution throughanescortdistribution is called theτ-transformation [16].
Foragiven pq(x;θ)∈Sq,wecandefinetheescortofanescortdistribution
P˜q(x;θ) :=Pq,(2)(x;θ) := q{Pq(x;θ)}q ′
= q{pq(x;θ)}2q−1.
We call P˜q(x;θ) the second escort distributionof pq(x;θ). The coefficient qbefore {pq(x;θ)}2q−1
comes from considerations ofU-information geometry [17]. Wewill discuss in the latter part of
Section5.
Similarly, we can define the n-th escort distribution Pq,(n)(x;θ) from the sequence of
escortdistributions:
Pq,(n)(x;θ) :={q(2q−1) · · ·((n−1)q−(n−2))}{pq(x;θ)}nq−(n−1). (5)
Let f(x)beafunctiononΩ. Theq-expectationEq,p[f(x)]andthenormalizedq-expectationEescq,p[f(x)]
withrespect to p(x;θ)∈Sq aredefinedby
Eq,p[f(x)] := ∫
Ω f(x)Pq(x;θ)dx,
Eescq,p[f(x)] := ∫
Ω f(x)Pescq (x;θ)dx,
respectively.Wedenoteby E˜q,p[f(x)] theexpectationwithrespect to thesecondescortdistribution
P˜q(x;θ), that is,
E˜q,p[f(x)] := ∫
Ω f(x)P˜q(x;θ)dx = q ∫
Ω f(x){pq(x;θ)}2q−1dx.
315
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik