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Friday, June 28, 2013

Solomonoff Universal Induction

In an earlier post (Kolmogorov, Solomonoff, and de Finetti) I linked to a historical article on the problem of induction. Here's an even better one, which gives a very clear introduction to Solomonoff Induction.
A Philosophical Treatise of Universal Induction
Samuel Rathmanner, Marcus Hutter

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.
Of course, properties such as Turing machine-independence and other key results are asymptotic in nature (only in the limit of very long sequences of data does it cease to matter exactly which reference Turing machine you choose to define program length). When it comes to practical implementations, the devil is in the details! You can think of the Solomonoff Universal Prior as a formalization of  the a priori assumption that the information in our Universe is highly compressible (i.e., there are underlying simple algorithms -- laws of physics -- governing its evolution). See also Information, information processing and black holes. From the paper:
... The formalization of Solomonoff induction makes use of concepts and results from computer science, statistics, information theory, and philosophy. It is interesting that the development of a rigorous formalization of induction, which is fundamental to almost all scientific inquiry, is a highly multi-disciplinary undertaking, drawing from these various areas. Unfortunately this means that a high level of technical knowledge from these various disciplines is necessary to fully understand the technical content of Solomonoff induction. This has restricted a deep understanding of the concept to a fairly small proportion of academia which has hindered its discussion and hence progress.

... Every major contribution to the foundations of inductive reasoning has been a contribution to under- standing rational thought. Occam explicitly stated our natural disposition towards simplicity and elegance. Bayes inspired the school of Bayesianism which has made us much more aware of the mechanics behind our belief system. Now, through Solomonoff, it can be argued that the problem of formalizing optimal inductive inference is solved.

Being able to precisely formulate the process of (universal) inductive inference is also hugely significant for general artificial intelligence. Obviously reasoning is synonymous with intelligence, but true intelligence is a theory of how to act on the conclusions we make through reasoning. It may be argued that optimal intelligence is nothing more than optimal inductive inference combined with optimal decision making. Since Solomonoff provides optimal inductive inference and decision theory solves the problem of choosing optimal actions, they can therefore be combined to produce intelligence. ... [ Do we really need Solomonoff? Did Nature make use of his Universal Prior in producing us? It seems like cheaper tricks can produce "intelligence" ;-) ]
Here are some nice informal comments by Solomonoff himself.

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