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SUMMARY:Probability paradigms in Uncertainty Quantification - Hermann Matt
 hies (Technische Universität Braunschweig)
DTSTART:20180117T110000Z
DTEND:20180117T130000Z
UID:TALK97972@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Probability theory was axiomatically built on the concept of m
 easure by A. Kolmogorov in the early 1930s\, giving the probability measur
 e and the related integral as primary objects and random variables\, i.e. 
 measurable functions\, as secondary. Not long after Kolmogorov&acute\;s wo
 rk\, developments in operator algebras connected to quantum theory in the 
 early 1940s lead to similar results in an approach where algebras of rando
 m variables and the expectation functional are the primary objects. Histor
 ically this picks up the view implicitly contained in the early probabilis
 tic theory of the Bernoullis.  This algebraic approach allows extensions t
 o more complicated concepts like non-commuting random variables and infini
 te dimensional function spaces\, as it occurs e.g. in quantum field theory
 \, random matrices\, and tensor-valued random fields. It not only fully re
 covers the measure-theoretic approach\, but can extend it considerably. Fo
 r much practical and numerical work\, which is often primarily concerned w
 ith random variables\, expections\, and conditioning\, it offers an indepe
 ndent theoretical underpinning. In short words\, it is &ldquo\;probability
  without measure theory&rdquo\;.  This functional analytic setting has str
 ong connections to the spectral theory of linear operators\, where analogi
 es to integration are apparent if they are looked for. These links extend 
 to the concept of weak distribution in a twofold way\, which describes pro
 bability on infinite dimensional vector spaces. Here the random elements a
 re represented by linear mappings\, and factorisations of linear maps are 
 intimately connected with representations and tensor products\, as they ap
 pear in numerical approximations.  This conceptual basis of vector spaces\
 , algebras\, linear functionals\, and operators gives a fresh view on the 
 concepts of expectation and conditioning\, as it occurs in applications of
  Bayes&acute\;s theorem. The problem of Bayesian updating will be sketched
  in the context of algebras via projections and mappings.  <br><br><br><br
 >
LOCATION:Seminar Room 2\, Newton Institute
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