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| In [[statistics]] and [[information geometry]], '''divergence''' or a '''contrast function''' is a function which establishes the “distance” of one [[probability distribution]] to the other on a statistical [[manifold]]. The divergence is a weaker notion than that of the [[distance]] in mathematics, in particular the divergence need not be symmetric (that is, in general the divergence from ''p'' to ''q'' is not equal to the divergence from ''q'' to ''p''), and need not satisfy the [[triangle inequality]].
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| == Definition ==
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| Suppose ''S'' is a space of all [[probability distribution]]s with common support. Then a '''divergence''' on ''S'' is a function {{nowrap|''D''(· {{!}}{{!}} ·): ''S×S'' → '''R'''}} satisfying <ref>{{harvtxt|Eguchi|1985}}</ref>
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| # ''D''(''p'' || ''q'') ≥ 0 for all ''p'', ''q'' ∈ ''S'',
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| # ''D''(''p'' || ''q'') = 0 if and only if ''p'' = ''q'',
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| # The matrix ''g''<sup>(''D'')</sup> (see definition in the “geometrical properties” section) is strictly [[Positive-definite matrix|positive-definite]] everywhere on ''S''.<ref>{{harvtxt|Amari|Nagaoka|2000|loc=chapter 3.2}}</ref>
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| The '''dual divergence''' ''D*'' is defined as
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| : <math>
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| D^*(p \parallel q) = D(q \parallel p).
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| </math>
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| == Geometrical properties ==
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| Many properties of divergences can be derived if we restrict ''S'' to be a statistical [[manifold]], meaning that it can be parametrized with a finite-dimensional coordinate system ''θ'', so that for a distribution {{nowrap|''p'' ∈ ''S''}} we can write {{nowrap|1=''p'' = ''p''(''θ'')}}.
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| For a pair of points {{nowrap|''p'', ''q'' ∈ ''S''}} with coordinates ''θ''<sub>''p''</sub> and ''θ''<sub>''q''</sub>, denote the partial derivatives of ''D''(''p'' || ''q'') as
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| : <math>\begin{align}
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| D((\partial_i)_p \parallel q) \ \ &\stackrel\mathrm{def}=\ \ \tfrac{\partial}{\partial\theta^i_p} D(p \parallel q), \\
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| D((\partial_i\partial_j)_p \parallel (\partial_k)_q) \ \ &\stackrel\mathrm{def}=\ \ \tfrac{\partial}{\partial\theta^i_p} \tfrac{\partial}{\partial\theta^j_p}\tfrac{\partial}{\partial\theta^k_q}D(p \parallel q), \ \ \mathrm{etc.}
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| \end{align}</math>
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| Now we restrict these functions to a diagonal {{nowrap|1=''p'' = ''q''}}, and denote <ref>{{harvtxt|Eguchi|1992}}</ref>
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| : <math>\begin{align}
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| D[\partial_i\parallel\cdot]\ &:\ p \mapsto D((\partial_i)_p \parallel p), \\
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| D[\partial_i\parallel\partial_j]\ &:\ p \mapsto D((\partial_i)_p \parallel (\partial_j)_p),\ \ \mathrm{etc.}
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| \end{align}</math>
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| By definition, the function ''D''(''p'' || ''q'') is minimized at {{nowrap|1=''p'' = ''q''}}, and therefore
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| : <math>\begin{align}
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| & D[\partial_i\parallel\cdot] = D[\cdot\parallel\partial_i] = 0, \\
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| & D[\partial_i\partial_j\parallel\cdot] = D[\cdot\parallel\partial_i\partial_j] = -D[\partial_i\parallel\partial_j] \ \equiv\ g_{ij}^{(D)},
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| \end{align}</math>
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| where matrix ''g''<sup>(''D'')</sup> is [[positive semi-definite]] and defines a unique [[Riemannian metric]] on the manifold ''S''.
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| Divergence ''D''(· || ·) also defines a unique [[torsion of connection|torsion]]-free [[affine connection]] ∇<sup>(''D'')</sup> with coefficients
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| : <math>
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| \Gamma_{ij,k}^{(D)} = -D[\partial_i\partial_j\parallel\partial_k],
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| </math>
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| and the [[dual affine connection|dual]] to this connection ∇* is generated by the dual divergence ''D''*. | |
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| Thus, a divergence ''D''(· || ·) generates on a statistical manifold a unique dualistic structure (''g''<sup>(''D'')</sup>, ∇<sup>(''D'')</sup>, ∇<sup>(''D''*)</sup>). The converse is also true: every torsion-free dualistic structure on a statistical manifold is induced from some globally defined divergence function (which however need not be unique).<ref>{{harvtxt|Matumoto|1993}}</ref>
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| For example, when ''D'' is an [[f-divergence]] for some function ƒ(·), then it generates the [[Riemannian metric|metric]] {{nowrap|1=''g''<sup>(''D''<sub>''f''</sub>)</sup> = ''c·g''}} and the connection {{nowrap|1=∇<sup>(''D''<sub>''f''</sub>)</sup> = ∇<sup>(''α'')</sup>}}, where ''g'' is the canonical [[Fisher information metric]], ∇<sup>(''α'')</sup> is the [[α-connection]], {{nowrap|1=''c'' = ƒ′′(1)}}, and {{nowrap|1=''α'' = 3 + 2ƒ′′′(1)/ƒ′′(1)}}.
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| ==Examples ==
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| The largest and most frequently used class of divergences form the so-called ''[[f-divergence]]s'', however other types of divergence functions are also encountered in the literature. | |
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| === f-divergences ===
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| {{Main|f-divergence}}
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| This family of divergences are generated through functions ''f''(''u''), convex on {{nowrap|''u'' > 0}} and such that {{nowrap|1=''f''(1) = 0}}. Then an ''f''-divergence is defined as
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| : <math>
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| D_f(p\parallel q) = \int p(x)f\bigg(\frac{q(x)}{p(x)}\bigg) dx
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| </math>
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| {| class="wikitable"
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| | [[Kullback-Leibler divergence]]:
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| | <math>
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| D_\mathrm{KL}(p \parallel q) = \int p(x)\ln\left( \frac{p(x)}{q(x)}\right) dx
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| </math>
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| |-
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| | squared [[Hellinger distance]]:
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| | <math>
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| H^2(p,\, q) = 2 \int \Big( \sqrt{p(x)} - \sqrt{q(x)}\, \Big)^2 dx
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| </math>
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| |-
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| | Jeffrey’s divergence:
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| | <math>
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| D_J(p \parallel q) = \int (p(x) - q(x))\big( \ln p(x) - \ln q(x) \big) dx
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| </math>
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| |-
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| | Chernoff’s [[α-divergence]]:
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| | <math>
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| D^{(\alpha)}(p \parallel q) = \frac{4}{1-\alpha^2}\bigg(1 - \int p(x)^\frac{1-\alpha}{2} q(x)^\frac{1+\alpha}{2} dx \bigg)
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| </math>
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| |-
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| | exponential divergence:
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| | <math>
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| D_e(p \parallel q) = \int p(x)\big( \ln p(x) - \ln q(x) \big)^2 dx
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| </math>
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| |-
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| | Kagan’s divergence:
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| | <math>
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| D_{\chi^2}(p \parallel q) = \frac12 \int \frac{(p(x) - q(x))^2}{p(x)} dx
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| </math>
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| |-
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| | (''α'',''β'')-product divergence:
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| | <math>
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| D_{\alpha,\beta}(p \parallel q) = \frac{2}{(1-\alpha)(1-\beta)} \int
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| \Big(1 - \Big(\tfrac{q(x)}{p(x)}\Big)^{\!\!\frac{1-\alpha}{2}} \Big)
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| \Big(1 - \Big(\tfrac{q(x)}{p(x)}\Big)^{\!\!\frac{1-\beta}{2}} \Big)
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| p(x) dx
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| </math>
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| |}
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| </ul>
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| === M-divergences ===
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| {{Empty section|date=January 2011}}
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| === S-divergences ===
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| {{Empty section|date=January 2011}}
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| == See also ==
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| * [[Statistical distance]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| * {{cite book
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| | last1 = Amari | first1 = Shun-ichi | authorlink = Shun'ichi Amari
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| | last2 = Nagaoka | first2 = Hiroshi
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| | title = Methods of information geometry
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| | year = 2000
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| | publisher = Oxford University Press
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| | isbn = 0-8218-0531-2
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| | ref = CITEREFAmariNagaoka2000
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| }}
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| * {{cite journal
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| | last = Eguchi | first = Shinto
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| | title = A differential geometric approach to statistical inference on the basis of contrast functionals
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| | year = 1985
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| | journal = Hiroshima mathematical journal
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| | volume = 15
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| | issue = 2
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| | pages = 341–391
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| | url = http://projecteuclid.org/euclid.hmj/1206130775
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| | ref = CITEREFEguchi1985
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| }}
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| * {{cite journal
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| | last = Eguchi | first = Shinto
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| | title = Geometry of minimum contrast
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| | year = 1992
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| | journal = Hiroshima mathematical journal
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| | volume = 22
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| | issue = 3
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| | pages = 631–647
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| | url = http://projecteuclid.org/euclid.hmj/1206128508
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| | ref = CITEREFEguchi1992
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| }}
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| * {{cite journal
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| | last = Matumoto | first = Takao
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| | title = Any statistical manifold has a contrast function — on the C³-functions taking the minimum at the diagonal of the product manifold
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| | year = 1993
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| | journal = Hiroshima mathematical journal
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| | volume = 23
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| | issue = 2
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| | pages = 327–332
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| | url = http://projecteuclid.org/euclid.hmj/1206128255
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| | ref = CITEREFMatumoto1993
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| }}
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| {{refend}}
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| {{DEFAULTSORT:Divergence (Statistics)}}
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| [[Category:Statistical distance measures]]
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| [[Category:F-divergences]]
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Break the difficulties of conformity as well as go all from a biking journey. Each hole is made up differently and your driver is not always the best way to start a hole. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. The remarkable thing though was an electric bike could cover 25 miles on one charge with a bet of pedal power or up to 15 miles on battery power lone. That kept me looking forward to the book launch and beyond.
The prices offered by these online stores are very competitive and realistic according to the market. One way to increase exercise fun is to find a friend to run with you. If you don't do a lot of racing on mountain trails then you need a road bike. Just to give you an idea, mine has three settings, those being; *Completely off, so that I am riding it as a normal bike; *Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). UST kits can also improve sturdiness of UST wheels with added sealing capacity in case of any punctures.
They have great suspension that fits the need of both amateur and professional bikers. That's a level of quality you just don't get from many brands these days. In case you have just about any issues about where by and also tips on how to utilize Womens mountain bike sizing., you possibly can call us at our own page. Buy a bike that has a top quality body made outside of steel, aluminum, carbon fiber or titanium. In no time you'll be cruising through the woods and getting into great shape. We have built a number of thousand semi custom wheels for all kinds of bikes, and will probably be happy to do the exact same for you.
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