Algorithmic Probability: Fundamentals and Applications

By Fouad Sabry

What Is Algorithmic Probability

In the field of algorithmic information theory, algorithmic probability is a mathematical method that assigns a prior probability to a given observation. This method is sometimes referred to as Solomonoff probability. In the 1960s, Ray Solomonoff was the one who came up with the idea. It has applications in the theory of inductive reasoning as well as the analysis of algorithms. Solomonoff combines Bayes' rule and the technique in order to derive probabilities of prediction for an algorithm's future outputs. He does this within the context of his broad theory of inductive inference.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Algorithmic Probability

Chapter 2: Kolmogorov Complexity

Chapter 3: Gregory Chaitin

Chapter 4: Ray Solomonoff

Chapter 5: Solomonoff's Theory of Inductive Inference

Chapter 6: Algorithmic Information Theory

Chapter 7: Algorithmically Random Sequence

Chapter 8: Minimum Description Length

Chapter 9: Computational Learning Theory

Chapter 10: Inductive Probability

(II) Answering the public top questions about algorithmic probability.

(III) Real world examples for the usage of algorithmic probability in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of algorithmic probability' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of algorithmic probability.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 28, 2023

Book preview

Chapter 1: Algorithmic probability

From observer states to physics via algorithmic probability[1]

Algorithmic probability, also known as Solomonoff probability, is a mathematical approach of assigning a prior probability to a given observation in the field of algorithmic information theory. In the 1960s, Ray Solomonoff came up with the idea. It finds application in the study of inductive reasoning and the evaluation of algorithms. Solomonoff incorporates Bayes' rule into his comprehensive theory of inductive inference to derive prediction probabilities for an algorithm's future outputs.

Finite binary strings representing the outputs of Turing machines serve as the observations in this mathematical framework, and the universal prior is a probability distribution over this set of strings derived from a distribution over programs (that is, inputs to a universal Turing machine). Since no string has a probability of zero, the prior is universal in the sense of Turing-computability. It cannot be computed exactly, but approximations can be made.

With the intention of applying it to machine learning, Solomonoff developed the theory of inductive inference, or prediction based on observations, which relies heavily on algorithmic probability: given a sequence of symbols, which one will appear next? Solomonoff's theory yields a solution that, while incomputable, is optimal in some sense. Solomonoff's theory is mathematically rigorous, in contrast to, say, Karl Popper's informal inductive reasoning theory.

Solomonoff's algorithmic probability can be traced back to these four primary sources. Applying Occam's razor, Epicurean theory of alternative explanations, current ideas in computer science (for.

use of a universal Turing machine) and Bayes’ rule for prediction.

Non-mathematical approximations of the universal prior include both Occam's razor and Epicurus' principle.

Occam's razor says to choose the simplest explanation among those that make sense given what we know about the world and what we've observed. As long as every computable function can be applied by at least one program on the abstract computer, we can consider it a Turing-complete machine.

The term simple explanation is given its specific meaning with the help of the abstract computer. In this formalism, theories of phenomena are like computer programs that, when executed on the abstract computer, produce strings of observations. The length of any computer program is used to calculate its relative importance. The universal probability distribution assigns the sum of the probabilities of the programs that compute something beginning with q as the probability distribution on all potential output strings with random input. Therefore, a brief piece of software is the simplest explanation. A lengthy piece of software represents a complex explanation. Since straightforward explanations are more likely to be correct, an observation string with a high probability is one that might have been generated by a relatively short computer program or by any of a huge number of slightly longer computer programs. It takes a lengthy computer program to construct an observation string with a low probability.

The idea of Kolmogorov complexity has direct ties to the field of algorithmic probability.

Complexity was first introduced by Kolmogorov due to issues in information theory and unpredictability, but Solomonoff had other motivations for introducing algorithmic complexity. logic based on inferences.

A single universal prior probability that can be substituted for each actual prior probability in Bayes’s rule was invented by Solomonoff with Kolmogorov complexity as a side product.

This restricts the amount of effort spent calculating potential program successes, with more time allotted to shorter shows.

When subjected to more prolonged operation, It will produce a series of estimates that eventually converge to the normal distribution of probabilities.

Other approaches to the problem involve reducing the scope of the search by employing training sequences.

Within a constant factor, as demonstrated by Solomonoff, this distribution is also machine-invariant (called the invariance theorem).

Around 1960, Solomonoff developed the idea of algorithmic probability and its associated invariance theorem, It is possible to concretize these concepts.

Ray Solomonoff

Andrey Kolmogorov

Leonid Levin

{End Chapter 1}

Chapter 2: Kolmogorov complexity

The Kolmogorov complexity of an object, like a piece of text, is defined as the length of the smallest computer program (in a fixed programming language) that generates the object as output in algorithmic information theory (a branch of computer science and mathematics). You may know it as algorithmic complexity, Solomonoff-Kolmogorov-Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy; all these names refer to the same thing: the amount of computer resources required to specify the item. It is a generalization of classical information theory and was initially published in 1963 under Andrey Kolmogorov's name.

Similar to Cantor's diagonal argument, the concept of Kolmogorov complexity can be utilized to express and prove impossible results, Gödel's incompleteness theorem, With the Turing-completeness paradox.

In particular, no program P computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than P's own length (see section § Chaitin's incompleteness theorem); Therefore, there is no way for a single program to accurately calculate the Kolmogorov complexity of an unlimited number of texts.

Take a look at the two sets of 32 numerals and lowercase letters below::

abababababababababababababababab , and

4c1j5b2p0cv4w1x8rx2y39umgw5q85s7

The first string is 17 characters long and represents a concise English description: write ab 16 times. The second string, consisting of 38 characters, cannot be described in any straightforwardly easy way (using the same character set) save from writing down the string itself, as in write 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7. Thus, it can be claimed that the first string-writing operation has less complexity than the second.

The formal definition of a string's complexity is the length of the shortest description of the string in a predetermined universal description language (the sensitivity of complexity relative to the choice of description language is discussed below). One may prove that the Kolmogorov complexity of any given string cannot be greater than a negligible multiple of the string's length in bytes. Strings like abab are not difficult because their Kolmogorov complexity is low compared to their size.

Although the Kolmogorov complexity can be defined for any mathematical object, for the sake of brevity we will just be discussing its application to strings. First, we need to choose on a string-description language. Any programming language, like Lisp, Pascal, or Java, might serve as the basis for such a description language. A description of the string x is P if and only if P is a program that produces x. Simply multiplying the length of P as a character string by the number of bits in a character yields the length of the description (e.g., 7 for ASCII).

A different option is to select an encoding for