Cyclic Directed Probabilistic Graphical Model: A Proposal Based on
  Structured Outcomes
Oleksii Sirotkin
In the process of building (structural learning) a probabilistic graphical
model from a set of observed data, the directional, cyclic dependencies between
the random variables of the model are often found. Existing graphical models
such as Bayesian and Markov networks can reflect such dependencies. However,
this requires complicating those models, such as adding additional variables or
dividing the model graph into separate subgraphs. Herein, we describe a
probabilistic graphical model - probabilistic relation network - that allows
the direct capture of directional cyclic dependencies during structural
learning. This model is based on the simple idea that each sample of the
observed data can be represented by an arbitrary graph (structured outcome),
which reflects the structure of the dependencies of the variables included in
the sample. Each of the outcomes contains only a part of the graphical model
structure; however, a complete graph of the probabilistic model is obtained by
combining different outcomes. Such a graph, unlike Bayesian and Markov
networks, can be directed and can have cycles. We explored the full joint
distribution and conditional distribution and conditional independence
properties of variables in the proposed model. We defined the algorithms for
constructing of the model from the dataset and for calculating the conditional
and full joint distributions. We also performed a numerical comparison with
Bayesian and Markov networks. This model does not violate the probability
axioms, and it supports learning from observed data. Notably, it supports
probabilistic inference, making it a prospective tool in data analysis and in
expert and design-making applications.