Modes and Maximum A Posteriori Estimators on Metric Spaces

Definitions and mathematical statement

Let $(X,d)$ be a metric space. Under the usual interpretation of the source notation $\mathcal P(X)$, let $\mathcal B(X)$ denote its Borel $\sigma$-algebra and let $\mu:\mathcal B(X)\to[0,1]$ be a countably additive measure with $\mu(X)=1$. The source does not explicitly specify the underlying $\sigma$-algebra; the Borel interpretation is therefore a convention requiring review. No completeness, separability, compactness, density, or reference measure is assumed in the supplied statement.

Research problem. Define and characterize mathematically meaningful notions of a point $x\in X$ being a most likely outcome of $\mu$, including notions of mode and maximum a posteriori (MAP) estimator. Determine hypotheses on the space and measure under which competing notions agree, and determine when a unique canonical notion can be distinguished.

For a specified family of competing definitions indexed by $I$, write $M_i(X,d,\mu)\subseteq X$ for the set of points satisfying definition $i$. Agreement then means \[ M_i(X,d,\mu)=M_j(X,d,\mu)\qquad(i,j\in I). \] This notation only expresses the comparison requested in the source: neither the family $I$ nor the defining criteria for $M_i$ are supplied. Likewise, the source gives no axioms defining what makes a notion canonical. Uniqueness of a notion must be distinguished from uniqueness of a point satisfying it; the latter would require a separate singleton condition. No posterior construction or statistical model is specified. Consequently, the supplied question is a research programme rather than a fully specified conjecture with a single truth value.

Short English statement

For a probability measure on a metric space, develop and compare definitions of a most likely point, including modes and MAP estimators. Identify conditions under which the definitions agree and a canonical definition can be justified. The source leaves the competing definitions and the meaning of canonical unspecified; uniqueness of a definition does not itself require a unique most likely point.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006262$, title "Modes and Maximum A Posteriori Estimators on Metric Spaces," source problem code catalogue entry.
• Source report: Uncertainty Quantification (2025), Oberwolfach Reports, pp. 1011--1078, DOI 10.4171/owr/2025/21. https://doi.org/10.4171/owr/2025/21. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Attribution in the original statement: "Defining Modes and MAP Estimators: A Systematic Approach," Ilja Klebanov, joint work with Hefin Lambley and T. J. Sullivan. The separate proposer field is unspecified.
• Supplied literature source, "OWR 2025 report": https://ems.press/content/serial-article-files/51850.
• Supplied literature source, "Classification of small-ball modes": https://arxiv.org/abs/2306.16278.
• Supplied literature source, "Lambley--Sullivan, MAP estimators": https://doi.org/10.1137/22M154243X.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that axiomatic work classifies multiple small-ball modes and MAP notions and identifies well-behaved cases, while there is no unique universal meaning of most likely for arbitrary metric-space measures. This is supplied dated curation, not independently verified here. The background explicitly calls for independent verification of time-sensitive status and cited claims.