Learning Recursive Functions Refutably

Learning of recursive functions refutably means that for every recursive function, the learning machine has either to learn this function or to refute it, i.e., to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. All these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are “most difficult” to learn, while the third one does. Moreover, we present characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general. For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. Then we derive hierarchies for refutable learning. Finally, we show that stricter refutability constraints cannot be traded for more liberal learning criteria

On Learning of Functions Refutably

Learning of recursive functions refutably informally means that for every recursive function, the learning machine has either to learn this function or to refute it, that is to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. Furthermore, all these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are “most difficult” to learn, while the third one does. Moreover, we present several characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general.For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. From this we derive some hierarchies for refutable learning. Finally, we prove that in general one cannot trade stricter refutability constraints for more liberal learning criteria

Low-Default Portfolio/One-Class Classification: A Literature Review

Consider a bank which wishes to decide whether a credit applicant will obtain credit or not. The bank has to assess if the applicant will be able to redeem the credit. This is done by estimating the probability that the applicant will default prior to the maturity of the credit. To estimate this probability of default it is first necessary to identify criteria which separate the good from the bad creditors, such as loan amount and age or factors concerning the income of the applicant. The question then arises of how a bank identifies a sufficient number of selective criteria that possess the necessary discriminatory power. As a solution, many traditional binary classification methods have been proposed with varying degrees of success. However, a particular problem with credit scoring is that defaults are only observed for a small subsample of applicants. An imbalance exists between the ratio of non-defaulters to defaulters. This has an adverse effect on the aforementioned binary classification method. Recently one-class classification approaches have been proposed to address the imbalance problem. The purpose of this literature review is three fold: (I) present the reader with an overview of credit scoring; (ii) review existing binary classification approaches; and (iii) introduce and examine one-class classification approaches