On the Identifiability of Finite Mixtures

Citation

Authors: Sidney Yakowitz, John D. Spragins
Year: 1968
Venue: The Annals of Mathematical Statistics 39(1), 209-214
arXiv: None
URL: https://doi.org/10.1214/aoms/1177698520

One-Line Takeaway

Yakowitz and Spragins give the clean structural criterion behind many later identifiability proofs: a family induces identifiable finite mixtures exactly when the generating cdfs are linearly independent.

Technical Summary

Setup: start with a family of cdfs $\mathcal{F} = \{F_\theta : \theta \in \Theta\}$ and ask when distinct finite signed combinations can agree pointwise.
Main theorem: the induced finite-mixture class is identifiable iff $\mathcal{F}$ is linearly independent, meaning $\sum_{j=1}^{m} c_j F_{\theta_j}(x) = 0$ for all $x$ implies $c_1 = \cdots = c_m = 0$ .
Consequence: later identifiability proofs can be reduced to verifying linear independence, and the paper checks this criterion for several classical families including multivariate Gaussians.

Assumptions And Limits

The criterion is conceptually broad but still requires verifying linear independence for the family at hand, which can be technically difficult. I could verify metadata and an abstract-level summary, but I could not retrieve a clean public full-text copy of the 1968 article during this session.

Relevance To This Project

This paper matters for the x -> z -> y story because it isolates the structural property that makes a latent decomposition recoverable from the observed law: distinct strategy-conditioned components must remain linearly independent in function space. If the candidate strategy family fails this test, then latent strategies are not identifiable from the marginal distribution alone.

Comparison Notes

Relative to Teicher (1963), this is the more abstract and reusable theorem. Relative to Chandra (1977), the emphasis is algebraic linear independence rather than convex geometry on measurable spaces.

Source Note

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

Verified from source: title, authors, venue, year, page range, and DOI were checked against bibliographic metadata for "On the Identifiability of Finite Mixtures," The Annals of Mathematical Statistics 39(1), 1968, 209-214, DOI 10.1214/aoms/1177698520; verification was limited to metadata and abstract-level claims because I did not recover a clean public full text during this session.