On the mixtures of probability distributions

Citation

Authors: Satish Chandra
Year: 1977
Venue: Scandinavian Journal of Statistics 4(3), 105-112
arXiv: None
URL: https://www.zentralblatt-math.org/zmath/en/search/?q=an:0389.62018

One-Line Takeaway

Chandra reframes finite-mixture identifiability at the level of measures and convex geometry, making clear that the primary estimand is the finitely supported mixing measure itself rather than a particular coordinate parameterization.

Technical Summary

Setup: on an arbitrary measurable space, take a family of laws $\mathcal{F} = \{P_\theta : \theta \in \Theta\}$ , a finitely supported mixing measure $G = \sum_{i=1}^{K} \alpha_i \delta_{\theta_i}$ , and the induced law $P_G = \sum_{i=1}^{K} \alpha_i P_{\theta_i}$ .
Central definition: finite-mixture identifiability is injectivity of the map $G \mapsto P_G$ on finitely supported probability measures over $\Theta$ .
Mathematical point: this formulation moves the problem from parameter tuples to convex geometry in the space of measures, separating identifiability of the mixing measure from identifiability of any chosen coordinates for its support atoms.

Assumptions And Limits

I could verify the bibliographic citation, but I did not find a clean public full-text copy or abstract for this 1977 article during the allotted retrieval pass. As a result, the content here is more interpretive than the other notes and should be treated as a project-facing reconstruction from how the paper is cited by later sources, not as a source-verified theorem digest.

Relevance To This Project

This is one of the clearest conceptual precursors for the repo's strategy-identification agenda. In the x -> z -> y framing, Chandra's viewpoint says that the mathematical object of interest is the mixing law over latent strategies, not the incidental coordinates used to describe those strategies. That distinction is useful whenever multiple parameterizations of the same latent behavior exist or when the component family is nonparametric.

Comparison Notes

Teicher (1963) and Yakowitz-Spragins (1968) are more theorem-forward. Chandra is the note to read for the geometric reformulation that later grouped-sample and nonparametric mixture work builds on.

Source Note

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Verified from source: the citation metadata Satish Chandra, "On the mixtures of probability distributions," Scandinavian Journal of Statistics 4 (1977), 105-112 was checked against bibliographic sources; verification did not extend to a public abstract or full text in this session.