  This is a short report on the discussions of appearance of tensors in
algebraic statistics and rigidity theory, during the semester ``AGATES:
Algebraic Geometry with Applications to TEnsors and Secants". We briefly survey
some of the existing results in the literature and further research directions.
We first provide an overview of algebraic and geometric techniques in the study
of conditional independence (CI) statistical models. We study different
families of algebraic varieties arising in statistics. This includes the
determinantal varieties related to CI statements with hidden random variables.
Such statements correspond to determinantal conditions on the tensor of joint
probabilities of events involving the observed random variables. We show how to
compute the irreducible decompositions of the corresponding CI varieties, which
leads to finding further conditional dependencies (or independencies) among the
involved random variables. As an example, we show how these methods can be
applied to extend the classical intersection axiom for CI statements. We then
give a brief overview about secant varieties and their appearance in the study
of mixture models. We focus on examples and briefly mention the connection to
rigidity theory which will appear in the forthcoming paper \cite{rigidity}.
