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2. Categorical Algorithmic Probability Classification.
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# Algorithmic probability

In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s. It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the prior obtained by this formula in Bayes' rule for prediction,

In the mathematical formalism used, the observations have the form of finite binary strings, and the universal prior is a probability distribution over the set of finite binary strings. The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated.

Algorithmic probability deals with the following questions:

• Given a body of data about some phenomenon that we want to understand, how can we select the most probable hypothesis of how it was caused from among all possible hypotheses?
• How can we evaluate the different hypotheses?
• How can we predict future data and how can we measure the likelihood of that prediction being the right one?

What inspired Solomonoff’s Algorithmic Probability?

Four principal inspirations for Solomonoff's algorithmic probability were:

• Occam's razor,
• Epicurus' principle of multiple explanations,
• Modern computing theory
• Bayes’ rule for prediction.

## What are the different types of algorithmic probability techniques?

### 1. An Algorithmic Probability Loss Function

The main task of a loss function is to measure the discrepancy between a value predicted by the model and the actual value as specified by the training data set. In most currently used machine learning paradigms this discrepancy is measured in terms of the differences between numerical values, and in case of cross-entropy loss, between predicted probabilities. Algorithmic information theory offers us another option for measuring this discrepancy–in terms of the algorithmic distance or information deficit between the predicted output of the model and the real value

### 2. Categorical Algorithmic Probability Classification

One of the main fields of application for automated learning is the classification of objects. These classification tasks are often divided into supervised and unsupervised problems.

### 3. Approximating the Algorithmic Similarity Function

While theoretically sound, the proposed algorithmic loss and classification cost functions rely on the uncomputable mathematical object K. However, recent research and the existence of increasing computing power have made available a number of techniques for the computable approximation of the non-conditional version of K.

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# Algorithmic probability

October 14, 2020

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In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s. It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the prior obtained by this formula in Bayes' rule for prediction,

In the mathematical formalism used, the observations have the form of finite binary strings, and the universal prior is a probability distribution over the set of finite binary strings. The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated.

Algorithmic probability deals with the following questions:

• Given a body of data about some phenomenon that we want to understand, how can we select the most probable hypothesis of how it was caused from among all possible hypotheses?
• How can we evaluate the different hypotheses?
• How can we predict future data and how can we measure the likelihood of that prediction being the right one?

What inspired Solomonoff’s Algorithmic Probability?

Four principal inspirations for Solomonoff's algorithmic probability were:

• Occam's razor,
• Epicurus' principle of multiple explanations,
• Modern computing theory
• Bayes’ rule for prediction.

## What are the different types of algorithmic probability techniques?

### 1. An Algorithmic Probability Loss Function

The main task of a loss function is to measure the discrepancy between a value predicted by the model and the actual value as specified by the training data set. In most currently used machine learning paradigms this discrepancy is measured in terms of the differences between numerical values, and in case of cross-entropy loss, between predicted probabilities. Algorithmic information theory offers us another option for measuring this discrepancy–in terms of the algorithmic distance or information deficit between the predicted output of the model and the real value

### 2. Categorical Algorithmic Probability Classification

One of the main fields of application for automated learning is the classification of objects. These classification tasks are often divided into supervised and unsupervised problems.

### 3. Approximating the Algorithmic Similarity Function

While theoretically sound, the proposed algorithmic loss and classification cost functions rely on the uncomputable mathematical object K. However, recent research and the existence of increasing computing power have made available a number of techniques for the computable approximation of the non-conditional version of K.