Download e-book for iPad: Uncertainty in artificial intelligence : proceedings of the by Ramon López de Mántaras; David L Poole

By Ramon López de Mántaras; David L Poole

ISBN-10: 1558603328

ISBN-13: 9781558603325

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Extra info for Uncertainty in artificial intelligence : proceedings of the Tenth Conference (1994) : July 29-31, 1994

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But it does not seem appropriate for her to do eitl1er; rather we would like to he able to generate a degree of belief in Cardinal(b) that takes into account the birdwatcher's degree of belief in Red( b ) . 8. 1 . A s another example, suppose we have reason to believe that two sensors are independent. For simplicity, suppose the sensors measure temperature, and report it to be either high, h, or low, /. We can imagine t11ree unary predicates: SJ ( i: ) , indicating that sensor 1 reports the value x; S2(x), a simi lar predicate for sensor 2; and A ctual( x ) , indicating that the actual temperature is x.

The application of equation (10) to approximate Var(g(0)jX) and Cov(g1 (0 ) , g 2 (0) jX) using the ex­ pressions (omitting 0 ) : .. 10 1 % error log scale Beta Posterior - Example �ost . 01 0 . ---'---< 10 20 3 0 40 50 60 70 80 90 100 n En ( 9192 I X) - En (g 1 ! X) ( l + O( n - 2 ) ) ( 14) and Figure 2 : Errors in Approximations for E(O I X) . ve ly, 0' 1 = p + +p+a +b a -2 . 0)7r(0) is always nonnegative) . This case can be addressed by at least two alternative approaches. The first one considers setting h(e, s) = exp(s g(0)) (that is al­ ways nonnegative) , computing Laplace's approxima­ tion for E( h ( e)) , the moment generating function ( mgf ) for g(0), fixing s at a convenient value where the mgf is defined, and then using the approximation & E( ,s)) E(g(0)) = ls= O (the definition of expecta­ tion from a mgf of a random variable).

Suppose that we want to compute the marginal posterior probability 7r(Bu IX) us­ ing Laplace 's method. In this example nep , neq L (Xj0) 7r (0) An approximation for equation ( 16) can be easily found for Bp = k using two alternative approaches. The first approach considers the use of Laplace's method to approximate both the integral part in equa­ tion ( 16) and the constant c defined in equation ( 1 7 ) . In the second approach the constant c i s approximated by an external procedure, usually numerical integra­ tion, that is very effective in low dimensions (and fre­ quently p = 1 or 2), according to Naylor and Smith [1982] .

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Uncertainty in artificial intelligence : proceedings of the Tenth Conference (1994) : July 29-31, 1994 by Ramon López de Mántaras; David L Poole


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