Identifiability of parameters in latent structure models with many observed variables

Citation

Authors: Elizabeth S. Allman, Catherine Matias, John A. Rhodes
Year: 2009
Venue: The Annals of Statistics 37(6A), 3099-3132
arXiv: arXiv:0809.5032
URL: https://doi.org/10.1214/09-AOS689

One-Line Takeaway

Allman, Matias, and Rhodes show that latent-class and related product-mixture models become generically identifiable once enough conditionally independent observed views are available and suitable rank conditions hold.

Technical Summary

Setup: latent-structure / latent-class models factor as $P(X_1,\ldots,X_p) = \sum_{h=1}^{r} \pi_h \prod_{j=1}^{p} P_{jh}(X_j),$ so after grouping observed variables into three blocks $S_1,S_2,S_3$ , the block law is a $3$ -way tensor with rank- $1$ components indexed by the latent class $h$ .
Core theorem: if the three block conditional-probability matrices $M_1,M_2,M_3$ have Kruskal ranks satisfying $\mathrm{krank}(M_1)+\mathrm{krank}(M_2)+\mathrm{krank}(M_3)\ge 2r+2,$ then the latent weights $\pi_h$ and blockwise conditional laws are identifiable up to permutation of the $r$ classes.
Main construction: for models with many observed variables, "clump" them into three conditionally independent super-variables, verify the resulting $M_i$ are generically full enough, and then apply Kruskal's uniqueness theorem for $3$ -way arrays.
Nonparametric extension: for mixtures of product measures with continuous coordinates, bin each coordinate to a finite partition, apply the same $3$ -block tensor argument to the binned variables, and lift the result back to the original measures under linear-independence conditions on the component cdfs.

Assumptions And Limits

The theorems are generic rather than universal, and they depend on conditional independence plus matrix/tensor rank conditions. They do not by themselves give a stable finite-sample algorithm or guarantee robustness near degenerate parameter configurations.

Relevance To This Project

This is one of the clearest papers for the repo's strategy-identification agenda because it shows how extra observed structure can restore identifiability even when single-view mixture decompositions are ambiguous. In x -> z -> y terms, multiple conditionally independent views of y can make latent strategies recoverable through tensor uniqueness rather than through restrictive component parametrization.

Comparison Notes

Compared with Yakowitz-Spragins (1968), the source of identifiability is not linear independence of a one-view component family, but multi-view tensor uniqueness. Arora et al. (2013) can be read as a more specialized and algorithmic descendant of this multi-view latent-variable line.

Source Note

Retrieval: alphaXiv MCP get_paper_content on arXiv:0809.5032 (returned full extracted paper text rather than a polished intermediate report); no local cache or downloaded source artifacts were created.

Verified from source: title, authors, year, DOI, arXiv identifier, venue, the $3$ -block Kruskal-rank condition, the clumping argument reducing richer latent-structure models to $3$ observed blocks, and the nonparametric binning-based identifiability construction were checked against the alphaXiv-extracted paper text.