# A Note on the Relationship Between Conditional and Unconditional Independence, and its Extensions for Markov Kernels

## TL;DR

This paper explores the relationship between conditional and unconditional independence, extending classical results to Markov kernels and providing new theorems, counterexamples, and representation results to clarify these concepts.

## Contribution

It introduces a main theorem linking independence of Markov kernels to conditional independence, extending existing results and providing new insights and counterexamples.

## Key findings

- Main theorem establishes minimal conditions for independence from conditional independence.
- Counterexamples clarify the boundaries of the theoretical results.
- Extensions to Markov kernels broaden the applicability of independence concepts.

## Abstract

Two known results on the relationship between conditional and unconditional independence are obtained as a consequence of the main result of this paper, a theorem that uses independence of Markov kernels to obtain a minimal condition which added to conditional independence implies independence. Some counterexamples and representation results are provided to clarify the concepts introduced and the propositions of the statement of the main theorem. Moreover, conditional independence and the mentioned results are extended to the framework of Markov kernels.