39 problems

Let $d\ge2$ and let $V_d^c$ be the scaled toric variety associated with the second hypersimplex, where $c\in(\mathbb{C}^*)^{d(d-1)/2}$ ranges over all scalings. Minimum ML degree c…

The ML-degree polynomiality conjecture. For generic $\mathcal{L}$, these ML-degrees are polynomial in $n$.

Let $X\subset\mathbb{C}\mathbb{P}^n$ be a closed irreducible subvariety with $X^\circ=X\setminus\mathcal{H}\neq\emptyset$. Its $i$-th sectional maximum likelihood degree is … where…

Consider generic sample values for an MA(2) likelihood problem, and distinguish its implicit formulation from its parametric formulation. The implicit formulation has an implicit M…

Let $n$ be the sample size, and let $(a_0,a_1)$ denote the MA(1) parameters. Restrict attention to solutions satisfying $\operatorname{Re}(a_0)>0$. Restricted MA(1) ML-degree conje…

Consider the poset of delta-matroid orbits ordered by inclusion, where each orbit arises from scaling matrices associated with a second hypersimplex. Monotonicity conjecture. The M…

Let $C_1,C_2\in\mathbb{C}^{d\times d}$ be scaling matrices, with associated delta-matroids, and let $c_1,c_2$ be the corresponding scalings. Delta-matroid invariance conjecture. If…

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

Let $\Delta_n$ be the model space with $n$ states. A model is reduced when it has no redundant reduction under the paper's model operations, and a reduced model is fundamental when…

A one-dimensional discrete model has a finite support, and its maximum likelihood estimator is the estimator obtained by maximizing the likelihood function from observed data; the…

Coons–Sullivant's monotonicity conjecture. The ML degree of a facial submodel cannot exceed the ML degree of the toric model itself; equivalently, ML degree is monotonic with respe…

Galois-width conjecture.

Let $C_n$ be the cycle graph on $n$ vertices, and let $\MLdeg(C_n)$ denote the number of critical points of the Gaussian log-likelihood function for generic sample covariance data,…

For the squared Grassmannian ${\rm sGr}(2,n)$, the maximum-likelihood degree is the number of complex critical points of the log-likelihood function for generic data. ML-degree con…

Small-ML-degree realization conjecture. The resulting scaled Segre embedding has ML degree $d$. This gives a proposed construction realizing every degree in the indicated range; th…

Let $w$ and $w'$ be scalings for the Segre embedding of $\mathbb P^1\times\mathbb P^1\times\mathbb P^{n}$, and let $P_n$ denote the relevant polytope. For every face…

Let $v$ and $w$ be scalings for $\mathbb P^{m-1}\times\mathbb P^{n-1}$, and define … Let $M_v$ and $M_w$ be the linear matroids defined by the columns of $\widehat v$ and…

Let $X_{A,w}$ be the scaled toric variety associated to a product of standard simplices, and let $m,n$ be positive integers. Universality conjecture. The result that every integer…

For integers $n$ and $d$ in the range where the corresponding linear covariance model is defined, let its ML-degree denote the number of critical points of the likelihood function…

Let $m$ denote the parameter indexing the associahedron, and consider toric statistical models supported on the associahedron. The minimal ML degree conjecture asserts that the num…

Let $A \in \mathbb{Z}^{n \times r}$ and $\overline{A} \in \mathbb{Z}^{n \times s}$ be a facial submatrix of $A$. The facial-submatrix maximum likelihood degree conjecture. The maxi…

Let $c6(n,a)$ denote the maximum likelihood degree for the symmetric matrix model with matrix size $n$ and parameter $a$. The authors' computational algorithms produce predictions…

Let $V$ be a vector space, let $\boldsymbol{S}^2V$ denote the space of symmetric matrices on $V$, and let $c6(n,a)$ be the maximum likelihood degree associated with symmetric matri…

Distinct-solutions conjecture. For generic values of $A$, $B$, and $S$, the score equations of $\mathcal{M}_{A,B}$ with sample covariance matrix $S$ have $2n-3$ distinct solutions.

Sturmfels–Timme–Zwiernik conjecture. The ML-degree in this case is $3n^2-9n+7$.