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\title{\bf Graph-Structure Discovery in a Logical Probabilistic Model}
\author{
    Greg Coppola \\
    {\em coppola.ai}
}
\date{\today}

\begin{document}
\maketitle

\section{Abstract}
A primary problem with large language models is that they ``hallucinate'' and that they ``do not reason''.
We have previously proposed \citep{coppola2024__theory_experiments} that both of these problems can be solved by adopting greater use of ``discrete, probabilistic methods'', in general, and what we have called a {\em Logical Bayesian Network} or a {\em Generative Logical Model}, in particular.

A {\em Logical Bayesian Network} shares with other {\em Bayesian Networks} the fact that the {\em structure} of the network is highly variable between domains and must be ``discovered'' as part of the learning process.
This gives another perspective on why the large language model has been successful.
A neural network can be viewed as a kind of (perhaps non-probabilistic) version of a graphical model in which the actual ``structure'' can be overly connected at first, with certain connections being differentiably ``learned to be ignored'' during training.
Historically, this must have been an easier problem to solve first than the non-differentiable, discrete structural growth required for a more traditional Bayesian Network to scale to an ``open domain''.

However, to obtain the already argued for benefits of a discrete logical generative model, we continue to want to pursue them.
Thus, we survey existing strategies and develop novel strategies for {\em growing} a discrete Bayesian Network, to fit the data.
We investigate strategies based on maximization of likelihood, including using expectation-maximization-based techniques, and the use of {\em hidden variables}.
We investigate the ``transfer'' of knowledge either from the large language model or from online information using {\em retrieval augmented generation}.
We also explore interpolations of these two strategies that we might call {\em hallucinate and test}—i.e., allowing the LLM to ``dream up'' new discrete connections, but then testing these against the data to see if the model really has ``improved''.

Finally, we remark on ways in which the structure discovery of a discrete logical model can mirror what is often referred to as ``creativity''.

\section{Dynamic Online Project}
The dynamically updated content—documents, code, and diagrams—for this project can be found at the {\em GitHub} project at \citep{coppola2025__discover_structure}.
We intend to package any applicable contributions as focused contributions to refereed papers.

\begin{thebibliography}{99}

\bibitem[Coppola, 2024]{coppola2024__theory_experiments}
Coppola, G. (2024).
\newblock The Quantified Boolean Bayesian Network: Theory and Experiments with
    a Logical Graphical Model.
\newblock arXiv preprint arXiv:2402.06557,
\newblock \url{https://arxiv.org/abs/2402.06557}.

\bibitem[Coppola, 2025]{coppola2025__discover_structure}
Coppola, G. (2025).
\newblock Graph-Structure Discovery in a Logical Probabilistic Model.
\newblock GitHub repository,
\newblock \url{https://github.com/gregorycoppola/discover-structure}.

\end{thebibliography}

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