Algorithmic probability

In his general theory of inductive inference, Solomonoff uses the method together with Bayes' rule to obtain probabilities of prediction for an algorithm's future outputs.

[3] In the mathematical formalism used, the observations have the form of finite binary strings viewed as outputs of Turing machines, and the universal prior is a probability distribution over the set of finite binary strings calculated from a probability distribution over programs (that is, inputs to a universal Turing machine).

Unlike, for example, Karl Popper's informal inductive inference theory,[clarification needed] Solomonoff's is mathematically rigorous.

Four principal inspirations for Solomonoff's algorithmic probability were: Occam's razor, Epicurus' principle of multiple explanations, modern computing theory (e.g. use of a universal Turing machine) and Bayes’ rule for prediction.

[5] Occam's razor and Epicurus' principle are essentially two different non-mathematical approximations of the universal prior.

The abstract computer is used to give precise meaning to the phrase "simple explanation".

[citation needed] Solomonoff's enumerable measure is universal in a certain powerful sense, but the computation time can be infinite.

Solomonoff proved this distribution to be machine-invariant within a constant factor (called the invariance theorem).

This corresponds to a scientists' notion of randomness and clarifies the reason why Kolmogorov Complexity is not computable.

Given that any uniquely-decodable code satisfies the Kraft-McMillan inequality, prefix-free Kolmogorov Complexity allows us to derive the Universal Distribution: where the fact that

[17] In terms of practical implications and applications, the study of bias in empirical data related to Algorithmic Probability emerged in the early 2010s.

The framework provides a foundation for creating universally intelligent agents capable of optimal performance in any computable environment.

It builds on Solomonoff’s theory of induction and incorporates elements of reinforcement learning, optimization, and sequential decision-making.

[23] Inductive reasoning, the process of predicting future events based on past observations, is central to intelligent behavior.

The framework is rooted in Kolmogorov complexity, which measures the simplicity of data by the length of its shortest descriptive program.

This concept underpins the universal distribution MM, as introduced by Ray Solomonoff, which assigns higher probabilities to simpler hypotheses.

Hutter extended the universal distribution to include actions, creating a framework capable of addressing problems such as prediction, optimization, and reinforcement learning in environments with unknown structures.

It describes a universal artificial agent designed to maximize expected rewards in an unknown environment.

AIXI operates under the assumption that the environment can be represented by a computable probability distribution.

It uses past observations to infer the most likely environmental model, leveraging algorithmic probability.

It computes their algorithmic probabilities and expected utilities, selecting the sequence of actions that maximizes cumulative rewards.

However, the general formulation of AIXI is incomputable, making it impractical for direct implementation.

AIXI is universally optimal in the sense that it performs as well as or better than any other agent in all computable environments.

However, its reliance on algorithmic probability renders it computationally infeasible, requiring exponential time to evaluate all possibilities.

To address this limitation, Hutter proposed time-bounded approximations, such as AIXItl, which reduce computational demands while retaining many theoretical properties of the original model.

The AIXI framework has significant implications for artificial intelligence and related fields.

It provides a formal benchmark for measuring intelligence and a theoretical foundation for solving various problems, including prediction, reinforcement learning, and optimization.

Additionally, its high computational requirements make real-world applications challenging.

The reliance on algorithmic probability ties intelligence to the ability to compute and predict, which may exclude certain natural or chaotic phenomena.

Nonetheless, the AIXI model offers insights into the theoretical upper bounds of intelligent behavior and serves as a stepping stone toward more practical AI systems.

From observer states to physics via algorithmic probability. A colorful illustration of Observation 5.4 shows the computational ontological model's possible histories, represented as a tree graph. The vertices correspond to the random variable (the computational state at time ), with valid states (black) forming a growing sequence of bit strings, while invalid states are shown in gray. Alice and Bob, observing an approaching meteorite, are part of the computational process, represented by and , respectively. Their shared emergent reality transitions probabilistically: with 99% likelihood, the meteorite hits Bob3rd, and with 1%, it misses. Postulates 3.2 imply this outcome is irrelevant for Bob1st, whose state transitions elsewhere, as generates a semimeasure, not a measure. The fate of Bob1st lies beyond the scope of Postulates 3.2, awaiting Postulates 3.1. [ 1 ]