V. Kreinovich, H.T. Nguyen, and E.A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control", In: K.M. Hanson and R.N. Silver, Eds., Maximum Entropy and Bayesian Methods, Dordrecht: Kluwer, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 34, 32, 40] if we are looking for the control that is the most robust in the average, then the best choice is to use f (a; b) a Delta b and f (a; b) a b Gamma a Delta b [25, 32, 35, 40] instead of minimizing the average error, we can try to minimize the corresponding entropy [4, 5, 13, 51, 52, 53]: if we use the average entropy (in some reasonable sense) we get the same pair of optimal functions as for average error; for an appropriately defined worst case entropy the optimal operations are f (a; b) min(a; b) and f (a; b) a b Gamma a Delta b. ffl If we are looking for the ....

.... to compute, then the best choice is to use f (a; b) min(a; b) and f (a; b) max(a; b) 16] ffl Finally, if, in control applications, we are looking for the most stable control for a given system, then the best choice is to use f (a; b) min(a; b) and f (a; b) a b Gamma a Delta b [13, 14, 15, 54]. These optimization results are in good accordance with the general group theoretic approach that enables us to classify techniques that are optimal relative to arbitrary reasonable criteria [1, 14, 15, 27, 54] This approach is a general approach to optimization under uncertainty. In practical ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver (eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996, pp. 93--100.

....to select an optimal formula for the so called inflation ; ffl in [18] to optimally select a sub box, and ffl in [22] for several other computational problems. Comment 2. We get the same uniform distribution as in Proposition 1 if we use a Maximum Entropy approach (see, e.g. 13] 17] [19]) i.e. select, among all possible probability distribution on the interval [a; b] a distribution with the largest possible entropy Gamma R ae(x) Delta log(ae(x) dx. This coincidence is not surprising, because the maximum entropy criterion is clearly shift and unit invariant. Comment 3. ....

V. Kreinovich, H.T. Nguyen, and E.A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control", In: K.M. Hanson and R.N. Silver, Eds., Maximum Entropy and Bayesian Methods, Dordrecht: Kluwer, 1996.

....an optimal formula for the so called inflation . In [21] we used symmetry to optimally select a sub box. In [25] we used symmetry for several other computational problems. Comment 2. We get the same uniform distribution as in Proposition 1 if we use a Maximum Entropy approach (see, e.g. [16, 20, 22]) i.e. select, among all possible probability distribution on the interval [a; b] a distribution with the largest possible entropy Gamma R ae(x) Delta log(ae(x) dx. This coincidence is not surprising, because the maximum entropy criterion is clearly shift and unitinvariant. Comment 3. ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver, Eds., Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 1996, pp. 93--100.

.... equivalence fj (a; b) we get fj (a; b) 1 Gamma a Gamma b 2 Delta a Delta b: It is worth mentioning that, in contrast to the expression for implication, expressions for exclusive or and for equivalence cannot be expressed as compositions of the algebraic operations and , or , and not [4]. Acknowledgments This work was supported in part by NASA grants NCC5 209 and NCC 2 1232, NSF grants CDA 9522207 and 9710940 Mexico Conacyt, Air Force Office of Scientific Research grant F49620 00 1 0365, and by Grant No. W 00016 from the U.S. Czech Science and Technology Joint Fund. The ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver (eds.), Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 1996, pp. 93--100.

.... similarity between the two formulas: it can be shown, e.g. that if we use the maximum entropy approach to select the most appropriate and and or operations, we get exactly the same operations f (a; b) a Delta b and f (a; b) a b Gamma a Delta b as our logic motivated approach [6]. There may also be a relation between the fact that the Maximum Entropy principle in statistics is often used to justify Gaussian distribution, and the fact that Gaussian membership functions are often used in fuzzy logic methodology (see, e.g. 4] and references therein; please note that there ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver (eds.), Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 1996, pp. 93--100.

.... similarity between the two formulas: it can be shown, e.g. that if we use the maximum entropy approach to select the most appropriate and and or operations, we get exactly the same operations f (a; b) a Delta b and f (a; b) a b Gamma a Delta b as our logic motivated approach [6]. There may also be a relation between the fact that the Maximum Entropy principle in statistics is often used to justify Gaussian distribution, and the fact that Gaussian membership functions are often used in fuzzy logic methodology (see, e.g. 4] and references therein; please note that there ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver (eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996, pp. 93--100.

.... fj (a; b) we get fj (a; b) 1 Gamma a Gamma b 2 Delta a Delta b: It is worth mentioning that, in contrast to the expression for implication, expressions for exclusive or and for equivalence cannot be expressed as compositions of the algebraic operations and , or , and not [4]. 6 Acknowledgments This work was supported in part by NASA under cooperative agreement NCC5 209, by the Future Aerospace Science and Technology Program (FAST) Center for Structural Integrity of Aerospace Systems, effort sponsored by the Air Force Office of Scientific Research, Air Force ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver (eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996, pp. 93--100.

....p i (x) There are many distribution of this type; we select the one for which the entropy Gamma P p(W ) Delta log(p(W ) takes the largest possible value. For this distribution, the desired probability of the disjunction is equal to p(x) 1 Gamma n Y i=1 (1 Gamma p i (x) see, e.g. [Kreinovich et al. 1996]) 5.2 Traditional Approach Is Not Always Sufficient From the statistical viewpoint, the MaxEnt formula corresponds to the case in which all modes are statistically independent. In reality, the detection errors in different modes can be caused by the same cause and therefore, these are not ....

Kreinovich, V., H. T. Nguyen, and E. A. Walker (1996) "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver, Eds., Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 93--100.

.... mathematical formulas) in [5] ffl The general idea of minimizing specificity is in line with such approaches as minimum specificity commitment principles in possibility theory (see, e.g. 4] or Dempster Shafer theory (see, e.g. 17] maximum entropy in probabilistic approach (see, e.g. [10, 14, 15, 16]) etc. 3 How to Elicit Numbers Without Asking for Them: Main Problems In the previous section, we formulated the idea of how to elicit numerical degrees of belief without explicitly asking for them. With this idea, come the following problems: ffl First of all, is there always a solution to ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: K. M. Hanson and R. N. Silver, Eds., Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996, pp. 93--100.

....the corresponding Lie transformation groups. As a result, we describe the optimal sensor placements. The efficiency of symmetry ideas is in line with the general efficiency of these ideas in computer science in general (see, e.g. 5] and in its maximum entropy problems in particular (see, e.g. [4]) First stage: Geometric techniques. The problem of choosing an optimal sensor placement is formulated in geometric terms: we need to select points (sensor placements) on a surface of the given structure. To solve this problem, we use the experience of solving similar geometric problems of ....

V. Kreinovich, H. T. Nguyen, and E. A. Walker, "Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations", In: Ken M. Hanson and Richard N. Silver, Eds., Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996, pp. 93--100.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC