24 problems

Let $C_n$ be the cycle with $n$ edges, and let $q(C_n)$ denote its consistency probability. Define … The cycle consistency probability conjecture. The consistency probability of…

Let $G=(V,A)$ be a directed acyclic graph, and let $\Sigma$ denote the covariance matrix of its arc-indicator variables. Covariance-matrix sparsity conjecture. The covariance matri…

Let $G=(V,A)$ be a directed acyclic graph, with arcs represented by indicator random variables $A_{ij}$ and covariance matrix $\Sigma$. Two arcs are incident when they share a comm…

Let $\mathcal{G}$ be a directed acyclic graph with skeleton $G$, and let $i$ be a vertex. Define $\mathcal{G}_{\downarrow i}$ by orienting every edge between $i$ and a neighbor tow…

Extremal ML threshold relation. For any undirected graph $G$,

Extremal ML degree bound. For any graph $G$,

Let $G$ be a random graph generated according to the Erdős–Rényi model $G(n,c/n)$, where $c$ is a fixed positive real number, and let $d$ be the minimum integer such that $G$ is…

Roverato's conjecture. For every real number $\delta>0$,

A BMT-derived graph is a graph obtained from the tree model by merging color classes of edges. Two internal nodes are non-adjacent when they are not joined by an edge in the underl…

Let $G$ be the undirected graphical model modelling a generic $n$-player game $X$ with binary choices in normal form. Let $\mathcal{C}=\operatorname{global}(G)$ and let…

Generic codimension additivity conjecture. If $\mathcal{S}_{G,\mathcal{X}}$ and all tent functions $h_{\mathcal{X}_C,y_C}$ are sufficiently generic, then

Support approximation conjecture. The sequence of sets converges to the MLE support:

Toric vanishing ideal conjecture. $I_{\mathcal{G}}$ is toric if and only if $\mathcal{G}$ is an RCOP block graph.

Coloured-cycle symmetry conjecture. All binomial linear forms in $I(\mathcal{L}^{-1})$ are induced by symmetries.

Let $G$ be a directed acyclic graph with node set $V$, and let $k>3$. For subsets $S_1,\ldots,S_k\subseteq V$ with … write $\mathcal{N}^{(k)}_{S_1,\ldots,S_k}$ for the subtensor of…

Let $G$ be an HTC-identifiable graph, let $\mathcal{M}(G)$ denote its Gaussian structural equation model, and let $J(\mathcal{M}(G))$ be the corresponding model ideal. The generato…

Dependently rigidity conjecture. The maximum likelihood threshold of $G$ is greater than $n$ if and only if $G$ is generically $n$-dependently rigid.

Lauritzen's conjecture. The graph $G$ is $n$-estimable if and only if

Sudderth–Wainwright–Willsky conjecture. If $f$ admits a pairwise, log-supermodular factorization over $G$, then

Let $G=(V,A)$ be a directed acyclic graph, and consider two arcs incident on a common node. Incident-arc range conjecture. The covariance between the two arcs takes values in the i…

Let $G=(V,A)$ be a directed acyclic graph, and consider two arcs that are incident on a common node. Let covariance and correlation refer to the corresponding arc-indicator random…

Monotonicity conjecture. Assume that $(G,d)$ does not have the positive margins (or interior point) property. If $d'_v\ge d_v$ for all $v\in V$, then $(G,d')$ does not have the pos…

Let $C$ be a cycle of length $n$, with edges $12, 23, \ldots, n1$. For a vertex $i$, let $d_i$ denote its associated model parameter, and let $\mu(C,d)$ be the Markov width of the…

Let $C_m$ be the $m$-cycle and let $P_{C_m}$ be the prime ideal of its Gaussian graphical model. Write $V(P_{C_m})$ for the associated projective variety. Degree formula conjecture…