"outside temperature" into a smaller number of intervals. Motivations There are several interrelated reasons for granulating variables in this fashion: Based on prior domain knowledge, there is no expectation that minute variations in temperature (e.g., the difference between ) could have an influence on behaviors driving the number of health-club applications. For this reason, any "regularity" which our learning algorithms might detect at this level of resolution would have to be spurious, as an artifact of overfitting. By coarsening the temperature variable into intervals the difference between which we do anticipate (based on prior domain knowledge) might influence number of health-club applications, we eliminate the possibility of detecting these spurious patterns. Thus, in this case, reducing resolution is a method of controlling overfitting. By reducing the number of intervals in the temperature variable (i.e., increasing its grain size), we increase the amount of sample data indexed by each interval designation. Thus, by coarsening the variable, we increase sample sizes and achieve better statistical estimation. In this sense, increasing granularity provides an antidote to the so-called curse of dimensionality, which relates to the exponential decrease in statistical power with increase in number of dimensions or variable cardinality. Independent of prior domain knowledge, it is often the case that meaningful regularities (i.e., which can be detected by a given learning methodology, representational language, etc.) may exist at one level of resolution and not at another. For example, a simple learner or pattern recognition system may seek to extract regularities satisfying a conditional probability threshold such as In the special case where this recognition system is essentially detecting logical implication of the form or, in words, "if then The system's ability to recognize such implications (or, in general, conditional probabilities exceeding threshold) is partially contingent on the resolution with which the system analyzes the variables. As an example of this last point, consider the feature space shown to the right. The variables may each be regarded at two different resolutions. Variable may be regarded at a high (quaternary) resolution wherein it takes on the four values or at a lower (binary) resolution wherein it takes on the two values Similarly, variable may be regarded at a high (quaternary) resolution or at a lower (binary) resolution, where it takes on the values or respectively. At the high resolution, there are no detectable implications of the form since every is associated with more than one