Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note on the Relationship Between Conditional and Unconditional
  Independence, and its Extensions for Markov Kernels   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.

25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an