## Conditional Independencies under the Algorithmic Independence of Conditionals

*Jan Lemeire*; 17(151):1−20, 2016.

### Abstract

In this paper we analyze the relationship between faithfulness
and the more recent condition of algorithmic Independence of
Conditionals (IC) with respect to the Conditional Independencies
(CIs) they allow. Both conditions have been extensively used for
causal inference by refuting factorizations for which the
condition does not hold. Violation of faithfulness happens when
there are CIs that do not follow from the Markov condition. For
those CIs, non-trivial constraints among some parameters of the
Conditional Probability Distributions (CPDs) must hold. When
such a constraint is defined over parameters of different CPDs,
we prove that IC is also violated unless the parameters have a
simple description. To understand which non-Markovian CIs are
permitted we define a new condition closely related to IC: the
Independence from Product Constraints (IPC). The condition
reflects that CIs might be the result of specific
parameterizations of individual CPDs but not from constraints on
parameters of different CPDs. In that sense it is more
restrictive than IC: parameters may have a simple description.
On the other hand, IC also excludes other forms of algorithmic
dependencies between CPDs. Finally, we prove that on top of the
CIs permitted by the Markov condition (faithfulness), IPC allows
non-minimality, deterministic relations and what we called
proportional CPDs. These are the only cases in which a CI
follows from a specific parameterization of a single CPD.

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