Identifiability of Finite Mixtures

Citation

Authors: Henry Teicher
Year: 1963
Venue: The Annals of Mathematical Statistics 34(4), 1265-1269
arXiv: None
URL: https://doi.org/10.1214/aoms/1177703862

One-Line Takeaway

Teicher gave one of the canonical early positive identifiability results for finite mixtures, proving that finite mixtures of normal and gamma families are identifiable and framing classical mixture recovery as a transform-based uniqueness problem.

Technical Summary

Setup: represent a finite mixture by a finitely supported mixing measure $G = \sum_{i=1}^{K} \alpha_i \delta_{\theta_i}$ and the induced law $P_G = \sum_{i=1}^{K} \alpha_i P_{\theta_i}$ .
Main result: for several classical families, most notably normal and gamma, equality $P_G = P_{G'}$ forces $G = G'$ up to permutation in minimal representations.
Proof idea: pass to a transform of the mixture, so the observed transform is a finite linear combination of component transforms; asymptotic comparison of distinct transform terms yields uniqueness.

Assumptions And Limits

The result is family-specific and finite-mixture-specific rather than a general criterion for arbitrary component classes. In this session I could verify bibliographic metadata directly, but I could not retrieve a clean public full-text copy of the 1963 article, so the technical content below is limited to abstract-level claims and later literature framing.

Relevance To This Project

For the repo's x -> z -> y framing, Teicher is the clean baseline case: if the strategy-conditioned family p(y | x, z) belongs to a sufficiently rigid parametric class, then recovering latent strategies from the marginal law is at least well-posed up to label permutation. The note is most relevant as a reminder that identifiability is not automatic; it has to come from restrictive structure in the component family.

Comparison Notes

Compared with Yakowitz and Spragins (1968), Teicher provides positive family-specific theorems rather than an if-and-only-if structural criterion. Compared with Chandra (1977), the paper is more concrete and less measure-theoretic.

Source Note

Retrieval: CAPES metadata page plus DOI/JSTOR metadata trail; no local cache or downloaded source artifacts were created.

Verified from source: title, author, venue, year, page range, and DOI were checked against DOI/JSTOR metadata for the original article; verification was limited to metadata and abstract-level claims because I did not recover a clean public full-text copy during this session.