Prediction and MDL for infinite sequences

Area of Science:

• Theoretical Computer Science
• Algorithmic Information Theory
• Machine Learning

Background:

• Solomonoff's universal prediction provides a theoretical framework for prediction based on Kolmogorov complexity.
• Algorithmic statistics offers methods for analyzing data using computable measures.
• Existing prediction methods may not guarantee bounded prediction errors for all data sequences.

Purpose of the Study:

• To develop a prediction method that combines the strengths of Solomonoff's universal prediction and algorithmic statistics.
• To ensure bounded prediction errors for observed data and along any Martin-Löf random sequence.
• To identify a computable measure that best explains observed data for enhanced predictive accuracy.

Main Methods:

• Integration of Solomonoff's universal prediction framework with principles of algorithmic statistics.
• Utilizing a computable measure that optimally "explains" observed data, as defined in algorithmic statistics.
• Analysis of prediction error bounds, specifically the sum of squares of prediction errors.

Main Results:

• The proposed method ensures that the expected sum of squares of prediction errors remains bounded.
• The method guarantees that the sum of squares of prediction errors is bounded along any Martin-Löf random sequence.
• Demonstration of a computable measure that provides superior data explanation for prediction.

Conclusions:

• The combined approach offers a theoretically sound and practically robust method for universal prediction.
• The findings advance the understanding of prediction bounds in the context of algorithmic information theory.
• This work provides a foundation for developing more reliable predictive models in machine learning and data analysis.