Belief Functions in Business Decisions
The book focuses on applications of belief functions to business decisions. Section I introduces the intuitive, conceptual and historical development of belief functions. Three different interpretations (the marginally correct approximation, the qualitati
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W. Liu
Hardware Implementation of Intelligent Systems
Propositional, Probabilistic and Evidential Reasoning
The book offers an ovuview of a large spectrum of implementations for th e computational intelligence based on neuro_fuzzy and artificia l approachu. The clear and concise explanat ions help the reader to understand the hardware implementation as~cts of the new computational intelligence paradigms.
Inl~raling Numerical and Symbolic Apploa.)g >-- >'12
12 in Fand A in
+ (1 -
(0,1),
>.)g.
Proposition 6.3 Continuity. For' alt J, g, k in F such that 1 >- k >-- g, there exist A, f.L in (0,1 J such that:
Al + (1
>.)g>-- k >- f.Ll
+ (1
f.L)g.
These three propositions are necessary and sufficient conditions for the existenee of a linear utility V on F representing ?:, Le., for the existence of V : F -+ R (where R is the set of re als ) satisfying;
V(f) ;::: V(g) and
V(f)
=
L
q
1 >- g.
V(lB)m(B)
B Z < - Y and we want to invert the edge Z < - Y, then we need to add the edge X - > Y. In case of general polytree we obtain in this way networks with considerably different undirected backbones:l. Ti we assurne that we adopt a structure (factorization) of Cano type, that is in form of a generalized polytree (singly connected "bayesian" network), then products of above transformations are directly convertible into a Shenoy's and Shafer's Markov tree. Let us now shift to Shenoy's and Shafer's belief propagation in Markov tree: The general principle there is "message passing" - if anode of Markov-tree gets information from its all but one neighbors , then it sends, to the remaining node, a "message", that is EIl combination of those messages plus its own factor of the belief function factorization. In the original Shenoy jShafer algorithm, this node's own factor of the belief function factorization is exactly the same independly to which neighbor the message is sent. We pro pose to have separate hypertrees for each target variable and to reason within each of them in one direction only (resp. modifications of propagation algorithm are known). Then it is guaranteed that the ff~sults of reasoning (a posteriori marginal distributions) will be marginally correct approximations with respect to the intrinsic distribution. Let us now consider the meaning of Cano et a1. conditionals. These are in fact sets of mappings between sets of variables selected with some prob ability. This gives a new meaning to the belief function. Instead of thinking in the way probability functions do that is that given some value of one variable, the conditional prob ability distribution assigns a value to another variable, we can think of objects that are assigned with some " In case of a general belief network transformations aiming at creation of directed pathes from every node towards the target variable would result in still more complicated networks so that merits of reasoning in such networks would he questionable
78
probability a belief.
2.4
Smnmary of Lower/Upper Bound Approximation
1. The case-based derivation of aposter
