Open Problems in Algebraic Statistics — General Problem 7

Definitions and mathematical statement

The supplied source asks to characterize the image of an entropy map $H$, its intersections with the faces of $\mathrm{SubMod}_n$, and its fibers. The record does not define the domain of $H$ or the precise convention for $\mathrm{SubMod}_n$.

Write $D$ for the intended domain and $C=\mathrm{SubMod}_n$ for the intended submodular cone. Once these objects and the map $H:D\to C$ are specified, the requested problems are to characterize \[ \operatorname{im}(H)=\{H(p):p\in D\}, \] to characterize, for each face $F$ of $C$, \[ \operatorname{im}(H)\cap F, \] and to describe, for each $h\in\operatorname{im}(H)$, the fiber \[ H^{-1}(h)=\{p\in D:H(p)=h\}. \] These expressions refer to the actual image; replacing it by its closure would require an additional qualification absent from the supplied statement.

For clarity, a possible finite discrete interpretation would take $D$ to be a specified family of joint probability laws of $n$ random variables and set \[ H(p)_S=-\sum_{x_S}p_S(x_S)\log_b p_S(x_S), \qquad \varnothing\ne S\subseteq\{1,\ldots,n\}, \] where $p_S$ is the marginal law, $b>1$, and $0\log_b0=0$, with $H(p)_\varnothing=0$ if the empty-set coordinate is included. A set function $h$ is submodular if \[ h(A)+h(B)\ge h(A\cup B)+h(A\cap B) \quad\text{for all }A,B\subseteq\{1,\ldots,n\}. \] This interpretation is conditional: the supplied record does not specify finite discrete variables, allowed alphabet sizes, the logarithm base, coordinate conventions, or whether $\mathrm{SubMod}_n$ also imposes normalization and monotonicity. Consequently, it does not determine a unique concrete entropy-map problem.

Short English statement

Characterize the entropy vectors attained by the intended map, determine which attained vectors lie on each face of the intended submodular cone, and describe all inputs producing a given entropy vector. The domain and cone conventions are missing from the supplied excerpt, so the concrete formulation requires review.

Sources

• Local UnsolvedMath registry: permanent ID $800868$, source problem code catalogue entry, title Open Problems in Algebraic Statistics --- General Problem 7. The supplied record attributes the problem to Sturmfels, with proposed year $2007$, and labels it open; that status is supplied registry metadata, not an independently verified assessment.
• Supplied source attribution: Sturmfels, Open Problems in Algebraic Statistics (2007), General Problem (#7), https://arxiv.org/abs/0707.4558. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The excerpt supplies the request concerning the image, face intersections, and fibers, but omits the definitions of its principal objects.