16 problems

Cubic generation conjecture. The prime ideal $Q_G$ is generated by the $3\times 3$-subdeterminants of any two-dimensional table obtained by flattening the $n$-dimensional table…

Let $\beta_1=|E|-|V|+c$ be the cycle rank (first Betti number) of the undirected skeleton of a directed acyclic graph, where $c$ is the number of connected components, and let…

Let a bitnet be a binary Bayesian network equipped with its Fisher information metric, and let $\langle R\rangle$ denote its volume-averaged Ricci scalar. Rodríguez's conjecture. F…

A finite Bayesian network is a statistical model associated with a directed acyclic graph and its ideal $P_G$; it is toric when the model can be realized as a toric variety, equiva…

Let $G$ be a finite Bayesian network, let $P_G$ be its associated prime ideal, and let $I_G$ be the ideal generated by the conditional-independence relations of $G$. The quadratic…

Expected-codimension and irreducibility conjecture. The variety $\mathcal{V}_{X,\mathcal{C}}$ has expected codimension $n$ inside $\mathcal{M}_\mathcal{C}$. It is positive-dimensio…

Sample complexity conjecture. The correct sample complexity for identity testing is

Consider a Bayesian network with source nodes $X$ (having no inbound edges) and sink nodes $Y$ (having no outbound edges). Select a node $Z$ and replace the channel from the parent…

Let $N$ be a finite set of nodes, let $\textsf{DAGS}(N)$ denote the acyclic directed graphs on $N$, and let $\mathsf{F}$ be the associated polytope whose graph vertices are…

Let $\mathcal{G}$ and $\mathcal{G}'$ be graphs, and write $\mathcal{G}\sim_m\mathcal{G}'$ when they induce the same model under the marginal Markov property, and…

Let agents receive private signals about a binary state and update their beliefs on the undirected chain according to the model considered in the paper, with agents' actions determ…

LP-relaxation conjecture. The polyhedron given by … should be an LP relaxation of the standard imset polytope.

LP-relaxation conjecture. The polyhedron given by those inequalities should be an LP relaxation of the standard imset polytope.

Gorenstein tree conjecture. The quotient ring $\mathbb{C}[\Sigma]/I_T$ is Gorenstein if and only if every vertex of $T$ has degree at most three.

Asymptotic scarcity conjecture. The proportion of DAGs on $n$ vertices satisfying

Normality and Cohen–Macaulayness conjecture. The coordinate ring of every Bayesian network is normal and Cohen–Macaulay.