A Riemannian Geometry in the q-Exponential Banach Manifold Induced by q-Divergences

28/08/2013
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A Riemannian Geometry in the q-Exponential Banach Manifold Induced by q-Divergences

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Objectives q-exponential statistical Banach manifold Induced Geometry manifold A Riemannian Geometry in the q-Exponential Banach Manifold Induced by q-Divergences Gabriel I. Loaiza O. Hector R. Quiceno Universidad EAFIT August 28, 2013 Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Table of Contents 1 Objectives 2 q-exponential statistical Banach manifold 3 Induced Geometry manifold Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Objectives Objective Give new mathematical developments to characterize the geometry induced by the q-exponential Banach manifold, by using q-divergence functionals, such that this geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Specific objectives 1 Introduce the metric. 2 Introduce the connections induced by the q-divergence functional, using the Eguchi relations. 3 Show the zero curvature of the manifold. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Objectives Objective Give new mathematical developments to characterize the geometry induced by the q-exponential Banach manifold, by using q-divergence functionals, such that this geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Specific objectives 1 Introduce the metric. 2 Introduce the connections induced by the q-divergence functional, using the Eguchi relations. 3 Show the zero curvature of the manifold. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Objectives Objective Give new mathematical developments to characterize the geometry induced by the q-exponential Banach manifold, by using q-divergence functionals, such that this geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Specific objectives 1 Introduce the metric. 2 Introduce the connections induced by the q-divergence functional, using the Eguchi relations. 3 Show the zero curvature of the manifold. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold Let q a real number such that 0 < q < 1. 1 We consider the q-deformed exponential and logarithmic functions which are respectively defined by ex q = (1 + (1 − q)x)1/(1−q) , if −1 1 − q ≤ x and lnq(x) = x1−q − 1 1 − q , if x > 0. 2 We consider the operations defined for real numbers x and y by x ⊕q y := x + y + (1 − q)xy and x q y := x − y 1 + (1 − q)y , for y = 1 q − 1 . 3 It holds that e (x1 q x2) q = ex q e x2 q and e (x1 q x2) q = ex q e x2 q . Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold Let q a real number such that 0 < q < 1. 1 We consider the q-deformed exponential and logarithmic functions which are respectively defined by ex q = (1 + (1 − q)x)1/(1−q) , if −1 1 − q ≤ x and lnq(x) = x1−q − 1 1 − q , if x > 0. 2 We consider the operations defined for real numbers x and y by x ⊕q y := x + y + (1 − q)xy and x q y := x − y 1 + (1 − q)y , for y = 1 q − 1 . 3 It holds that e (x1 q x2) q = ex q e x2 q and e (x1 q x2) q = ex q e x2 q . Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold Let (Ω, Σ, µ) be a probability space. Denote by Mµ the set of strictly positive probability densities µ-a.e. For each p ∈ Mµ consider the probability space (Ω, Σ, p · µ), where (p · µ)(A) = A pdµ. Denote · p,∞ the norm in L∞(p · µ), with which Bp := {u ∈ L∞ (p · µ) : Ep[u] = 0}, is a closed normed subspace so is a Banach space. The probability densities p, z ∈ Mµ are connected by a one-dimensional q-exponential model if there exist r ∈ Mµ, u ∈ L∞(r · µ), a real function of real variable ψ and δ > 0 such that for all t ∈ (−δ, δ), the function f defined by f (t) = e tu q ψ(t) q r, satisfies that there are t0, t1 ∈ (−δ, δ) for which p = f (t0) and z = f (t1). The function f is called one-dimensional q-exponential model. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold We define the mapping Mp given by Mp(u) = Ep[e (u) q ], denoting its domain by DMp ⊂ L∞(p · µ). Also we define the mapping Kp : Bp,∞(0, 1) → [0, ∞], for each u ∈ Bp,∞(0, 1), by Kp(u) = lnq[Mp(u)]. If restricting Mp to Bp,∞(0, 1), this function is analytic and infinitely Fréchet differentiable. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold Let (Ω, Σ, µ) be a probability space and q a real number with 0 < q < 1. Let be Vp := {u ∈ Bp : u p,∞ < 1}, for each p ∈ Mµ. We define the maps eq,p : Vp → Mµ by eq,p(u) := e (u q Kp(u)) q p, which are injective and their ranges are denoted by Up. For each p ∈ Mµ the map sq,p : Up → Vp given by sq,p(z) := lnq z p q Ep lnq z p , is precisely the inverse map of eq,p. Maps sq,p are the coordinate maps for the manifold and the family of pairs (Up, sq,p)p∈Mµ define an atlas on Mµ; and the transition maps, for each u ∈ sq,p1 (Up1 Up2 ), are given by sp2 (ep1 (u)) = u ⊕q lnq( p1 p2 ) − Ep2 [u ⊕q lnq( p1 p2 )] 1 + (1 − q)Ep2 [u ⊕q lnq( p1 p2 )] . Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold q-exponential statistical Banach manifold Given u ∈ sq,p1 (Uq,p1 Uq,p2 ), we have that D(sq,p2 ◦ s−1 q,p1 )(u) · v = A(u) − B(u)Ep2 [A(u)], where A(u), B(u) are functions depending on u. The collection of pairs {(Up, sq,p)}p∈Mµ is a C∞-atlas modeled on Bp, and the corresponding manifold is called q−exponential statistical Banach manifold. Finally, the tangent bundle of the manifold, is characterized, (Proposition 15), by regular curves on the manifold, where the charts (trivializing mappings) are given by (g, u) ∈ T (Up) → (sq,p(g), A(u) − B(u)Ep[A(u)]), defined in the collection of open subsets Up × Vp of Mµ × L∞(p · µ). Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold The q-divergence functional is given as follow. Let f be a function, defined for all t = 0 and 0 < q < 1, by f (t) = −t lnq 1 t and for p, z ∈ Mµ. The q-divergence of z with respect to p is given by I(q) (z||p) := Ω p f z p dµ = 1 1 − q 1 − Ω zq p1−q dµ , (1) which is the Tsallis divergence functional. We have that the manifold is related with the q-divergence functional as sq,p(z) = 1 1 + (q − 1)I(q)(p||z) lnq z p + I(q)(p||z) . Proposition Let p, z ∈ Mµ then (du)z I(q)(z||p)|z=p = (dv )p I(q)(z||p)|z=p = 0, where the subscript p, z means that the directional derivative is taken with respect to the first and the second arguments in I(q)(z||p), respectively, along the direction u ∈ Tz (Mµ) or v ∈ Tp(Mµ). Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold The q-divergence functional is given as follow. Let f be a function, defined for all t = 0 and 0 < q < 1, by f (t) = −t lnq 1 t and for p, z ∈ Mµ. The q-divergence of z with respect to p is given by I(q) (z||p) := Ω p f z p dµ = 1 1 − q 1 − Ω zq p1−q dµ , (1) which is the Tsallis divergence functional. We have that the manifold is related with the q-divergence functional as sq,p(z) = 1 1 + (q − 1)I(q)(p||z) lnq z p + I(q)(p||z) . Proposition Let p, z ∈ Mµ then (du)z I(q)(z||p)|z=p = (dv )p I(q)(z||p)|z=p = 0, where the subscript p, z means that the directional derivative is taken with respect to the first and the second arguments in I(q)(z||p), respectively, along the direction u ∈ Tz (Mµ) or v ∈ Tp(Mµ). Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold According to Proposition (16), the functional I(q)(z||p) is bounded, since: I(q)(z||p) ≥ 0 and equality holds iff p = z and I(q)(z||p) ≤ Ω (z − p) f z p dµ. Then, together with previous proposition, the q-divergence functional induces a Riemannian metric g and a pair of connections, see Eguchi, given by: g(u, v) = −(du)z (dv )p I(q) (z||p)|z=p (2) w u, v = −(dw )z (du)z (dv )p I(q) (z||p)|z=p, (3) where v ∈ Tp(Mµ), u ∈ Tp(Mµ) and w is a vector field. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold Denote Σ(Mµ) the set of vector fields u : Up → Tp(Up), and F(Mµ) the set of C∞ functions f : Up → R. The following result establish the metric. By direct calculation over I(q)(z||p), we obtain (dv )pI(q)(z||p) = 1 1−q Ω (1 − q) − (1 − q)p(−q) z(q) vdµ and (du)z (dv )p I(q)(z||p) = −q Ω p(−q) z(q−1) uvdµ, so by (2), it follows g(u, v) = q Ω uv p dµ. Ten we have the follow result. Proposition Let p, z ∈ Mµ and v, u vector fields, the metric tensor (field) g : Σ(Mµ) × Σ(Mµ) → F(Mµ) is given by g(u, v) = q Ω uv p dµ. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold Denote Σ(Mµ) the set of vector fields u : Up → Tp(Up), and F(Mµ) the set of C∞ functions f : Up → R. The following result establish the metric. By direct calculation over I(q)(z||p), we obtain (dv )pI(q)(z||p) = 1 1−q Ω (1 − q) − (1 − q)p(−q) z(q) vdµ and (du)z (dv )p I(q)(z||p) = −q Ω p(−q) z(q−1) uvdµ, so by (2), it follows g(u, v) = q Ω uv p dµ. Ten we have the follow result. Proposition Let p, z ∈ Mµ and v, u vector fields, the metric tensor (field) g : Σ(Mµ) × Σ(Mµ) → F(Mµ) is given by g(u, v) = q Ω uv p dµ. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold Proposition The connections are characterized as follows. The family of covariant derivatives (connections) (q) w u : Σ(Mµ) × Σ(Mµ) → Σ(Mµ), is given as (q) w u = dw u − 1 − q p uw. It is easy to prove that the associated conjugate connection is given by ∗(q) w u = dw u − q p uw. Notice that taking q = 1−α 2 yields to the Amaris’s one-parameter family of α−connections in the form (α) w u = dw u − 1 + α 2p uw; and taking q = 1 2 the Levi-Civita connection results. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold Proposition The connections are characterized as follows. The family of covariant derivatives (connections) (q) w u : Σ(Mµ) × Σ(Mµ) → Σ(Mµ), is given as (q) w u = dw u − 1 − q p uw. It is easy to prove that the associated conjugate connection is given by ∗(q) w u = dw u − q p uw. Notice that taking q = 1−α 2 yields to the Amaris’s one-parameter family of α−connections in the form (α) w u = dw u − 1 + α 2p uw; and taking q = 1 2 the Levi-Civita connection results. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold Induced Geometry manifold Proposition Finally, we characterize this geometry by calculating the curvature and torsion tensors, for which it will be proved that equals zero, i.e, for the q-exponential manifold and the connection given in the previous proposition, the curvature tensor and the torsion tensor satisfy R(u, v, w) = 0 and T(u, v) = 0. Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold References Amari, S.: Differential-geometrical methods in statistics. Springer, New York (1985) Amari, S., Nagaoka, H.: Methods of information Geometry. RI: American Mathematical Society. Translated from the 1993 Japanese original by Daishi Harada, Providence (2000) Amari, S. Ohara, A.: Geometry of q-exponential family ofprobability distributions. Entropy. 13, 1170-1185 (2011) Borges, E.P.: Manifesta˜oes dinˆamicas e termodinˆamicas de sistemas n˜ao-extensivos. Tese de Dutorado, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro (2004). Cena, A., Pistone, G.: Exponential statistical manifold. Annals of the Institute of Statistical Mathematics. 59, 27-56 (2006) Dawid, A.P: On the conceptsof sufficiency and ancillarity in the presence of nuisance parameters. Journal of the Royal Statistical Society B. 37, 248-258 (1975) Efron, B.: Defining the curvature of a statistical problem (with applications to second order efficiency). Annals of Statistics. 3, 1189-1242 (1975) Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold References Eguchi, S.: Second order efficiency of minimum coontrast estimator in a curved exponential family. Annals of Statistics. 11, 793-803 (1983) Furuichi, S.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45, 4868-4877 (2004) Gibilisco, P., Pistone, G.: Connections on non-parametric statistical manifolds by Orlicz space geometry. Infinite Dimensional Analysis Quantum Probability and Related Topics. 1, 325-347 (1998) Kadets, M.I., Kadets, V.M: series in Banach spaces, Birkaaauser Verlang, Besel. Conditional and undconditional convergence, Traslated for the Russian by Andrei Iacob. (1997). Kulback, S., Leibler, R.A.: On Information and Sufficiency. Annals of Mathematics and Statistics. 22, 79-86 (1951) Loaiza, G., Quiceno, H.R.: A q-exponential statistical Banach manifold. Journal of Mathematical Analysis and Applications. 398, 446-476 (2013). Pistone, G.: k-exponential models from the geometrical viewpoint. The European Physical Journal B. Springer Berlin. Online 70 29-37 (2009) Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. The Annals of statistics. 23(5), 1543–1561 (1995). Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J.Stat. Phys. 52, 479-487 (1988)Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold Objectives q-exponential statistical Banach manifold Induced Geometry manifold THANKS Gabriel I. Loaiza O. Hector R. Quiceno A Riemannian Geometry in the q-Exponential Banach Manifold