Aleshin-Guendel 2020 - Identifiability of Latent Class MSE Models - Overview

Summary

Aleshin-Guendel (2020, arXiv:2008.09865) proves that a family of $J$-class latent class models for multiple-systems estimation with $K$ lists is identifiable if and only if $2J\le K$. Applications of Manrique-Vallier’s (2016) LCMCR method that use more classes than $K/2$ (e.g. the default $J^{\ast }=5$ with fewer than 10 lists) therefore use non-identifiable families.

Overview

Why this paper stands in for Manrique-Vallier (2016). Manrique-Vallier’s Biometrics paper (“Bayesian population size estimation using Dirichlet process mixtures”) is only available from the author’s university site, which blocks automated downloads (not circumvented). This arXiv paper is a legal open source that restates the latent-class MSE model, states the identification condition, and shows what is wrong with the LCMCR default. LCMCR itself is covered in Concept - Latent Class and Dirichlet Process Models for Multiple Systems Estimation (concept note, no raw source). Manrique-Vallier (2016) remains a follow-on ingest.

Contributions. (1) A mechanism for verifying identifiability in $M_{th}$ models via moments of the mixing distribution (Theorem 1). (2) Necessary and sufficient condition for latent class models (Theorem 2): $2J\le K$.

Key numbers. $K=2$, $J=2$: two distinct two-class models give the same observed cell probabilities $\tilde{\pi }=(0.276,0.276,0.448)$ but missing-cell probabilities $0.316$ vs $0.219$, i.e. different population sizes for the same data (Table 1).

Motivating PyMC examples

From https://redam94.github.io/pymc-challenges/examples.html:

• E109 - “Counting the uncounted” (latent-class MSE with classes summed out and label switching; Kosovo via LCMCR).
• E69 - “Truth without a gold standard: latent class models for noisy raters and diagnostic tests” (parameter vs degrees-of-freedom counting; the same latent-class identification logic).

Source Map

Paper section	Covered in
Abstract, 1 Introduction (Mh, Mth, identifiability literature)	this note; Aleshin-Guendel 2020 - Mth Model and Identifiability Definition
2.1 Model description; 2.2 identifiability in Mth models (Def. 1); 2.3 latent class models	Aleshin-Guendel 2020 - Mth Model and Identifiability Definition
3 Mixed moments (Lemma 1, Theorem 1)	Aleshin-Guendel 2020 - Mixed-Moments Theorem
4.1 The answer (Theorem 2, proof)	Aleshin-Guendel 2020 - Latent Class Identifiability Theorem and LCMCR
4.2 Implications (LCMCR default, Table 1, applications)	Aleshin-Guendel 2020 - Latent Class Identifiability Theorem and LCMCR
Appendix 1 (full rank), Appendix 2 (counterexample moments)	Aleshin-Guendel 2020 - Latent Class Identifiability Theorem and LCMCR
References	folder index (Sources section)

Connections

• Empirical instability of LCMCR on real data: Silverman 2020 - Madigan-York and Dirichlet Process Benchmarks.
• N-mixture identifiability by assumption, not by data: Barker 2018 - Reliability of N-Mixture Models - Overview.
• Mixture models background: Monsters and Mixtures, Nonparametric Models Overview.

See Also

• Concept - Closed-Population Capture-Recapture and Otis Models
• Concept - Unseen Species and Latent Class Links