# Decomposition of conditional probability for high-order symbolic Markov chains

## TL;DR

This paper introduces a decomposition method for the conditional probability function of high-order symbolic Markov chains, enabling better sequence modeling by capturing complex correlations.

## Contribution

It develops a novel decomposition procedure representing the conditional probability as a sum of multi-linear memory functions, bridging likelihood estimation and additive Markov chains.

## Key findings

- Memory functions are expressed in terms of high-order correlations at weak dependencies.
- The method allows sequential sequence generation considering increasing correlation complexity.
- Potential application in neural network training approximation.

## Abstract

The main goal of the paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear memory function monomials of different orders (from zero up to the chain order). This allows us to construct artificial sequences by method of successive iterations taking into account at each step of iterations increasingly more high correlations among random elements. At weak correlations, the memory functions are uniquely expressed in terms of the high-order symbolic correlation functions. The proposed method fills up the gap between two approaches: the likelihood estimation and the additive Markov chains. The obtained results might be used for sequential approximation of artificial neural networks training.