On the complexity of hypothesis space and the sample complexity for machine learning

The problem of learning a concept from examples in the model introduced by Valiant is discussed. According to the traditional ways of thinking, it is assumed that the learnability is independent of the occurrence probability of instance. By utilizing this probability, we propose the metric as a new measure to determine the complexity of hypothesis space. The metric measures the hardness of discrimination between hypotheses. Furthermore, we obtain the average metric dependent on prior information. This metric is the measure of complexity for hypothesis space in the average. Similarly in the worst case, we obtain the minimum metric. We make clear the relationship between these measures and the Vapnik - Chervonenkis (VC) dimension. Finally, we show the upper bound on sample complexity utilizing the metric. This results can be applied in the discussion on the learnability of the class with an infinite VC dimension.