Download Information Algebras: Generic Structures For Inference by Juerg Kohlas PDF

By Juerg Kohlas

Information often is available in items, from assorted assets. It refers to various, yet similar questions. for that reason details has to be aggregated and centred onto the correct questions. contemplating mixture and focusing of data because the appropriate operations ends up in a general algebraic constitution for info. This publication introduces and experiences info from this algebraic standpoint. Algebras of data give you the precious summary framework for conventional inference systems. they permit the applying of those techniques to a wide number of diverse formalisms for representing info. whilst they enable a frequent research of conditional independence, a estate regarded as basic for wisdom presentation. details algebras offer a normal framework to outline and examine doubtful details. doubtful info is represented through random variables that clearly shape info algebras. This conception additionally pertains to probabilistic assumption-based reasoning in details structures and is the foundation for the assumption features within the Dempster-Shafer conception of evidence.

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The rest follows from (6), since x, y ~ xU y. xn y )1y. x and 4> represent the same knowledge. 2) Extending a valuation to a larger domain and marginalizing back should give, if not the initial valuation, then at least a valuation equivalent to the initial one. 3) if x ~ y . 3) is called a stable congruence. Note for example, that proportionality of probability potentials (see above) defines a stable congruence on the valuation algebra of probability potentials. 44 3. 2). From now on we consider stable valuation algebras in this section.

WI. If we introduce this into the second This is again a relation onto which we have to condition the normal distribution of W. Also, clearly, in this case the focal manifolds of the two hints intersect into one-dimensional manifolds (if they intersect). So the combined hint becomes a precise Gaussian hint. 85) = (In, (A' . ~-l . A)-I. A' . ~-l . 1, (A' . ~-l . A)-I). We remark that for example the combination of two precise hints leads to this case. After some simple algebraic transformations, one can verify that this rule of combination is identical to the one defined for Gaussian potentials above.

For all sED there is an element es with d(e s ) with d( ¢) = s, es 0 ¢ = ¢ 0 e s = ¢ . = s such that for all ¢ E <1> 2. Labelzng: For ¢, 'lj; E <1>, d( ¢ 0 'lj;) = d( ¢) U d( 'lj;). 88) 3. 89) 4. 90) 5. 91) 6. 92) As usual, we define marginalization of ¢ E <1> by ¢It = ¢-CdCcP)-t) , if d(¢) - t E V(¢). Let M(¢) = {t : d(¢) - t E V(¢) . 93) It is like the usual transitivity axiom, when the marginalization to t is possible. The (extended) combination axiom, in terms of marginalization, becomes: let d(¢) = x, d('lj;) = y, x s;;: z s;;: xUy.

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