Understanding inductive reasoning is a problem that has engaged mankind for thousands of years. This problem is relevant to a wide range of fields and is integral to the philosophy of science. It has been tackled by many great minds ranging from philosophers to scientists to mathematicians, and more recently computer scientists. In this article we argue the case for Solomonoff Induction, a formal inductive framework which combines algorithmic information theory with the Bayesian framework. Although it achieves excellent theoretical results (...) and is based on solid philosophical foundations, the requisite technical knowledge necessary for understanding this framework has caused it to remain largely unknown and unappreciated in the wider scientific community. The main contribution of this article is to convey Solomonoff induction and its related concepts in a generally accessible form with the aim of bridging this current technical gap. In the process we examine the major historical contributions that have led to the formulation of Solomonoff Induction as well as criticisms of Solomonoff and induction in general. In particular we examine how Solomonoff induction addresses many issues that have plagued other inductive systems, such as the black ravens paradox and the confirmation problem, and compare this approach with other recent approaches. (shrink)

Practical reasoning requires decision—making in the face of uncertainty. Xenelda has just left to go to work when she hears a burglar alarm. She doesn’t know whether it is hers but remembers that she left a window slightly open. Should she be worried? Her house may not be being burgled, since the wind or a power cut may have set the burglar alarm off, and even if it isn’t her alarm sounding she might conceivably be being burgled. Thus Xenelda can (...) not be certain that her house is being burgled, and the decision that she takes must be based on her degree of certainty, together with the possible outcomes of that decision. (shrink)

Using formal systems to represent and reason about uncertainty.