FUZZY INDEPENDENCE AND EXTENDED CONDITIONAL-PROBABILITY

In many applications, the use of Bayesian probability theory is proble matical. Information needed to feasibility calculate is unavailable. T here are different methodologies for dealing with this problem, e.g., maximal entropy and Dempster-Shafer Theory. If one can make independen ce assumptions, many of the problems disappear, and in fact, this is o ften the method of choice even when it is obviously incorrect. The not ion of independence is a 0-1 concept, which implies that human guesses about its validity will not lead to robust systems. In this paper, we propose a fuzzy formulation of this concept. It should lend itself to probabilistic updating formulas by allowing heuristic estimation of t he ''degree of independence.'' We show how this can be applied to comp ute a new notion of conditional probability (we call this ''extended c onditional probability''). Given information, one typically has the ch oice of full conditioning (standard dependence) or ignoring the inform ation (standard independence). We list some desiderata for the extensi on of this to allowing degree of conditioning. We then show how our fo rmulation of degree of independence leads to a formula fulfilling thes e desiderata. After describing this formula, we show how this compares with other possible formulations of parameterized independence. In pa rticular, we compare it to a linear interpolant, a higher power of a l inear interpolant, and to a notion originally presented by Hummel and Manevitz [Tenth Int. Joint Conf. on Artificial Intelligence, 1987]. In terestingly, it turns out that a transformation of the Hummel-Manevitz method and our ''fuzzy'' method are close approximations of each othe r. Two examples illustrate how fuzzy independence and extended conditi onal probability might be applied. The first shows how linguistic prob abilities result from treating fuzzy independence as a linguistic vari able. The second is an industrial example of troubleshooting on the sh op floor.