By Juerg Kohlas
Information often is available in items, from varied assets. It refers to diverse, yet comparable questions. for that reason info has to be aggregated and targeted onto the appropriate questions. contemplating blend and focusing of knowledge because the correct operations results in a popular algebraic constitution for info. This e-book introduces and reports details from this algebraic perspective. Algebras of data give you the useful summary framework for established inference techniques. they enable the applying of those tactics to a wide number of assorted formalisms for representing details. even as they enable a general examine of conditional independence, a estate regarded as basic for wisdom presentation. info algebras supply a traditional framework to outline and examine doubtful details. doubtful details is represented by way of random variables that evidently shape info algebras. This concept additionally pertains to probabilistic assumption-based reasoning in details platforms and is the foundation for the idea capabilities within the Dempster-Shafer idea of evidence.
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Additional info for Information Algebras: Generic Structures For Inference
Then we have >lxuy = ((>TxuY)lr)lxuy, 'lj;Txuy = (('Ij;TxuY)Tr)lxuy by stability. So we obtain >Txuy = (>Tr)lxuy = ('Ij;Tr)lxuy = 'lj;Txuy. If > == 'Ij; (mod 0"), then stability implies also, if d(» = x,d('Ij;) = y, that ('Ij; TXUY)lx = >, (> lxUY)1Y = 'Ij;. 9) We can thus start with the valuation 'Ij;, extend it to xU y, marginalize to x to obtain > and vice versa. So, equivalent valuations modulo 0" really represent the same information. 1 (8) we have also > --+Y = (>lxny) lY .
Although marginals of normalized m-functions are still normalized, this is not true in general for the combination. Thus under the actual definition of combination (Mn, D) is not a sub-valuation algebra of (M, D), it is not a valuation algebra at all. 30 2. Valuatwn Algebras But there is an alternative definition of combination for normalized mfunctions. 73) AIsu'nAi su ',e0 Furthermore we define ml ® m2(0) = O. If K = 0, then, ml ® m2(A) = 0 for all A i= 0 and ml ® m2(0) = 1. This way to combine scaled set potentials is called Dempster's rule (Shafer, 1976) .
Xny) Txuy But this says that ¢> =="IjJ (mod a). 13) is itself a remarkable result. xny , that is the two equivalent valuations are identical on their largest common domain. The equivalence relation a is now shown to be a congruence relative to the operations of combination and transport. 4 1. If ¢>1 =="ljJ1 (mod a) and ¢>2 =="ljJ2 (mod a), then ¢>1 <9 ¢>2 == "ljJ1 <9"IjJ2 (mod a). 14) 2. If ¢> =="IjJ (mod a), then for all zED, ¢>--+z == "IjJ-+z (mod a). 15) Proof. (1) Let Xl, X2, Y1, Y2 be the domains of ¢>1, ¢>2, "ljJ1,"ljJ2 respectively.