Probabilistic Learning of Indexed Families under Monotonicity Constraints -- Hierarchy Results and Complexity Aspects

We are concerned with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider con-servative, strong-monotonic and monotonic probabilistic learning of indexed families with respect to class comprising, class preserving and proper hypothesis spaces, and investigate the probabilistic hierarchies in these learning models. In the setting of learning indexed families, probabilistic learning under monotonicity constraints is more powerful than deterministic learning under monotonicity constraints, even if the probability is close to 1, provided the learning machines are restricted to proper or class preserving hypothesis spaces. In the class comprising case, each of the investigated probabilistic hierarchies has a threshold. In particular, we can show for class comprising conservative learning as well as for learning without additional constraints that probabilistic identification and team identification are equivalent. This yields discrete probabilistic hierarchies in these cases. In the second part of our work, we investigate the relation between probabilistic learn