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UID:www.tcs.tifr.res.in/event/581
DTSTAMP:20230914T125930Z
SUMMARY:Statistical Inference Based on a Parametric Family of Divergences
DESCRIPTION:Speaker: M. Ashok Kumar (Indian Statistical Institute\nTheorect
ical Statistics and Mathematics Unit\n8th Mile\, Mysore Road\nRVCD Post\nB
angalore 560059)\n\nAbstract: \nAbstract: We study minimization problems w
ith respect to a one-parameter family of generalized divergences (denoted
$I_{\\alpha}(P\,Q)$). These $I_{\\alpha}$-divergences are a generalization
of the so-called Kullback-Leibler divergence (KL-divergence). Just like K
L-divergence\, these $I_{\\alpha}$-divergences also behave like squared Eu
clidean distance and satisfy the Pythagorean property. This talk is about
the usefulness of these geometric properties in robust statistics. The tal
k is organized in three parts.\n\nIn the first part\, we study minimizatio
n of $I_{\\alpha}(P\,Q)$ as the first argument varies over a family of pro
bability distributions that satisfy linear statistical constraints. Such a
constraint set is called a linear family. This minimization problem gener
alizes the maximum Renyi or Tsallis entropy principle of statistical physi
cs. The structure of the minimizing probability distribution naturally sug
gests a statistical model of power-law probability distributions\, which w
e call an $\\alpha$-power-law family. This is analogous to the exponential
family that arises when relative entropy is minimized subject to the same
linear statistical constraints.\n\nIn the second part\, we study minimiza
tion of $I_{\\alpha}(P\,Q)$ over the second argument. This minimization is
generally on parametric families such as the exponential family or the $\
\alpha$-power-law family\, and is of interest in robust statistical estima
tion.\n\nIn the third part\, we show an orthogonality relationship between
an $\\alpha$-power-law family and an associated linear family. As a conse
quence of this\, the minimization of $I_{\\alpha}$ over the second argumen
t on an $\\alpha$-power-law family can be shown to be equivalent to a mini
mization of $I_{\\alpha}$ over the first argument on a linear family. The
latter turns out to be a simpler problem of minimization of a quasi-convex
objective function subject to linear constraints (this is a joint work wi
th Rajesh Sundaresan).\n
URL:https://www.tcs.tifr.res.in/web/events/581
DTSTART;TZID=Asia/Kolkata:20150302T113000
DTEND;TZID=Asia/Kolkata:20150302T123000
LOCATION:D-405 (D-Block Seminar Room)
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