Inheritance of Statistical Properties

Statistical aggregate properties (e.g. mean, maximum, mode) have not previously been thought to “inherit” between sets. But they do in a weak sense, and a collection of such “weak” information can be combined in a rule-based architecture to get stronger information. I I Motivation S~~ppose we have conducted a census of all elephants in the world and we can definitely say that all elephants are gray. Then by set-to-subset inheritance of the “color” property, the set of elephants in Clyde’s herd must be gray, Clyde’s herd being some particular herd of elephants. This will not work for statistical aggregate properties such as maximum and mean. Suppose our census found that the longest elephant in the world is 27 feet long, and the average elephant 15 feet. This does not mean the longest elephant in Clyde’s herd is 27 feet, nor the average in the herd 15 feet. But a weak form of inheritance is present, for we can assign different degrees of Iikelihood to the following: This work is part of the Knowledge Base Management Systems Project, under contract ## N00039-82-G-0250 from the Defense Advanced Research Projects Agency of the United States Department of Defense. The views and conclusions contained in this dcrcument are those of the author and should not be interpreted as representative of the official policies of DARPA or the US Government. I. “The longest elephant in Clyde’s herd is 30 feet long.” 2. “The average elephant in Clyde’s herd is 30 feet long.” 3. “The longest elephant in Clyde’s herd is 27 feet long.” 4. “The average elephant in Clyde’s herd is 27 feet long.” 5. “The longest elephant in Clyde’s herd is 16 feet long.” 6. “The average elephant in Clyde’s herd is 16 feet long.” Statements 1 and 2 are impossible. Statement 3 is possible but a bit unlikely, whereas statement 4 is almost certainly impossible. Statement 5 is surprising and hence apparently unlikely, whereas 6 is quite reasonable. Since we don’t know anything of Clyde’s herd other khan that they are elephants, a kind of inheritance from the properties of elephants in general must be happening. The issue here is more important than elephants. Thousands of databases in existence s~tpport statistical questions about their contents. Exact answers to such questions may be very time-consuming for large data sets and/or remote access. Many users, especially nonstatisticians, may be willing instead to accept much faster approximate answers via inheritance methods [5]. 221 From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved. 2. Our four-characteristic approach We wish to address inheritance of the set properties maximum, mean, standard deviation, median, mode, fits to simple distributions, and correlations between different values of the same item. Our theory concerns set representation only (but sets of cardinality one can represent individuals). It concerns “definitional” sets primarily (those with absolute criteria for membership) as opposed to “natural kind” sets [l] (though degrees of set membership as in fuzzy set theory could be introduced). The theory mainly deals with extensions (exemplars), not intensions (meanings). It also only addresses the setsubset semantic relationship; however, often other relationships can be viewed this way by “atomization” of the included concepts, e.g. geographical containment may be seen as a set-subset relationship between sets of points. The key is to note that while in a few cases statistical properties inherit values exactly from set to set, in most cases they do not; but that there are characterizations of a numeric statistic that will inherit much more often: o an upper bound on its value l a lower bound on its value 8 a best estimate of the value o a standard deviation of possibilities for the value