A Philosophical Treatise of Universal Induction (1105.5721v1)

Abstract: 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.

• The paper’s main contribution is advancing Solomonoff induction with a Bayesian framework to resolve traditional challenges in inductive reasoning.
• It unifies philosophical, historical, and technical insights by revisiting ideas from Hume, Ockham, and Sextus Empiricus.
• It examines the computational limits of Solomonoff induction and highlights practical approximations like the MDL principle for real-world applications.

An Analysis of "A Philosophical Treatise of Universal Induction"

Samuel Rathmanner and Marcus Hutter's "A Philosophical Treatise of Universal Induction" provides a comprehensive and multidisciplinary explanation of Solomonoff induction. This paper explores the historical, philosophical, and technical aspects of inductive reasoning, ultimately presenting Solomonoff's method as a promising framework for understanding and implementing universal induction.

At its core, the paper advances Solomonoff induction within the Bayesian framework. This approach combines inductive inference with algorithmic probability, a concept introduced by Ray Solomonoff. The result is a method that aims to resolve the challenges associated with traditional inductive reasoning systems, such as the notorious black ravens paradox and the confirmation problem. Despite its strong theoretical foundation, Solomonoff induction remains relatively obscure in the wider scientific community due to its technical complexity.

Solomonoff induction addresses key issues like the black ravens paradox by positing that any enough sequence of similar events supports these types of universal generalizations. The use of algorithmic probability allows for a resolution of Hume's problem of induction, as it treats all computable hypotheses as possible and assigns them non-zero prior probabilities. This is a stark shift from traditional Bayesian approaches where exact hypotheses could end up with zero prior probability, thereby failing to be confirmed by evidence regardless of its weight, thus resolving the zero prior problem.

The authors devote significant attention to the philosophical underpinnings of induction, tracking its historical trajectory through figures like Sextus Empiricus, David Hume, and William of Ockham. These discussions illuminate how Solomonoff induction naturally extends these philosophical foundations into a formal system that promises to unify various understandings of induction under a single theoretical umbrella. Especially interesting is the paper’s assertion that Solomonoff induction provides the theoretical advances that earlier figures like Hume and Ockham could only muse about.

While the foundational propositions of Solomonoff induction hold great appeal, the authors acknowledge the inherent challenges it faces, particularly its incomputability, which makes practical application challenging. Despite this, the authors argue that it serves as an ideal or "gold standard" against which practical systems might be measured. Like other "gold standards" in scientific inquiry (quantum electrodynamics in physics, for instance), Solomonoff induction provides an asymptotic target for real-world implementations, even if it remains theoretically out of reach due to its computational requirements.

Several approximations and related approaches are discussed that attempt to capitalize on Solomonoff’s theoretical ideals while remaining computationally feasible. The Minimum Description Length (MDL) principle is highlighted as an approximate method rooted in similar philosophical tenets that employs practical data compression techniques as surrogates for Kolmogorov complexity. Such approximations have been effectively utilized in data prediction and machine learning, notably through well-regarded methods such as context tree weighting algorithms.

In conclusion, Rathmanner and Hutter’s treatise stands as an exhaustive exploration of Solomonoff induction, positioning it as a theoretically robust yet computationally demanding framework for solving the problem of universal induction. The essay advocates for the continued paper and approximation of Solomonoff’s methods, suggesting that advancements in computation and algorithmic techniques might increasingly allow Solomonoff induction to inform the future of artificial intelligence and computational learning. As a theoretical concept, it reorients the perspective on how we address the induction problem, proposing a more unified and philosophically grounded approach. However, the paper acknowledges the practical limitations that necessitate ongoing research and development for effective implementation in real-world systems.