Aleshin-Guendel 2020 - Mth Model and Identifiability Definition

Summary

$K$ sources sample a closed population of unknown size $N$; each individual has latent sampling probabilities ${\lambda }_i\sim Q$ and is included by source $k$ with probability ${\lambda }_{ik}$ independently given ${\lambda }_i$. The observed cell counts are multinomial and the likelihood factors into a binomial for $n$ and a conditional multinomial; identifiability means the conditional cell probabilities determine the missing-cell probability.

Overview

Classical models (Otis et al. 1978): $M_h$ (heterogeneity; same latent probability for every source) and $M_{th}$ (heterogeneity and source effects). $M_h$ can be plausible in ecology where animals can be given equal catchability; in human-population settings sources are convenience samples so $M_{th}$ is needed. Manrique-Vallier (2016) proposed latent class $M_{th}$ models with many classes, now widely used in human-rights work (El Salvador, Sri Lanka, Palembang, Peru, Colombia, Philippines, Kampala, South Sudan). Identifiability of $M_h$ families has literature (Huggins 2001; Link 2003; Holzmann et al. 2006), but not $M_{th}$.

Main Content

Setup and $M_{th}$ model (Sec. 2.1, Eq. 1)

$K\ge 2$ sources; $H=\{0,1{\}}^K$ inclusion patterns; $H^{\ast }=H\setminus \{0{\}}^K$ patterns of the $n<N$ individuals sampled at least once; $x_i\in H$ the pattern for individual $i$; $n_h={\sum }_{i=1}^{N}I(x_i=h)$ cell counts of a $2^K$ table; $n_{(0,\dots ,0)}=N-n$ is unobserved and $N$ is the target.

$$x_{ik}\mid {\lambda }_{ik}\stackrel{ind}{\sim }\text{Bernoulli}({\lambda }_{ik}),({\lambda }_{i1},\dots ,{\lambda }_{iK})\stackrel{iid}{\sim }Q,Q\in \mathcal{Q},$$

with $\mathcal{Q}$ a family of mixing distributions on $(0,1{]}^K$ (every individual has non-zero probability of being sampled by at least one source). $M_h$ is the sub-model with ${\lambda }_{i1}=\cdots ={\lambda }_{iK}$.

Likelihood and its decomposition (Eq. 2-3)

Marginalising over $Q$, the full table is multinomial:

$$(n_h{)}_{h\in H}\mid N,{\pi }_Q\sim \text{Multinomial}(N,{\pi }_Q),{\pi }_{Q,h}=E_Q\{\prod _{k=1}^K{\lambda }_k^{h_k}(1-{\lambda }_k{)}^{1-h_k}\}.$$

Equivalently,

$$n\mid N,{\pi }_{Q,0}\sim \text{Binomial}(N,1-{\pi }_{Q,0}),(n_h{)}_{h\in H^{\ast }}\mid n,{\tilde{\pi }}_Q\sim \text{Multinomial}(n,{\tilde{\pi }}_Q),$$

with ${\pi }_{Q,0}={\pi }_{Q,(0,\dots ,0)}$ and ${\tilde{\pi }}_{Q,h}={\pi }_{Q,h}/(1-{\pi }_{Q,0})$. The second factor is the conditional likelihood (Fienberg 1972). To estimate $N$, the conditional probabilities ${\tilde{\pi }}_Q$ estimated from observed cells must determine the missing-cell probability ${\pi }_{Q,0}$.

Definition 1 (identifiability of a mixing family)

A family $\mathcal{Q}$ of distributions on $(0,1{]}^K$ is identifiable if, for $Q,R\in \mathcal{Q}$, ${\tilde{\pi }}_Q={\tilde{\pi }}_R$ implies ${\pi }_{Q,0}={\pi }_{R,0}$. (Modified from Link 2003.) If so, $N$ can be consistently estimated within $\mathcal{Q}$ (Sanathanan 1972).

Latent class family ${\mathcal{Q}}_J$ (Sec. 2.3)

$${\mathcal{Q}}_J=\{Q=\sum _{j=1}^J{\nu }_{Q,j}\prod _{k=1}^K{\delta }_{{\lambda }_{Q,jk}}:{\nu }_{Q,j}\ge 0,\sum _j{\nu }_{Q,j}=1,{\lambda }_{Q,jk}\in (0,1{]}^K\}.$$

Class $j$ has weight ${\nu }_{Q,j}$ and class-specific sampling probabilities ${\lambda }_{Q,j\cdot }$; this is a finite mixture of product Bernoullis (Goodman 1974; Haberman 1979). Whether ${\mathcal{Q}}_J$ is identified was unknown.

Parameter count: $J(K+1)-1$ in ${\mathcal{Q}}_J$. The conditional probabilities ${\tilde{\pi }}_Q$ have $2^K-2$ free dimensions, so identifiability requires at least $J(K+1)-1\le 2^K-2$. The paper shows the true condition is far stricter.

Examples

$K=2$ sources: cell probabilities

$H=\{00,01,10,11\}$ and $H^{\ast }=\{01,10,11\}$, so $\tilde{\pi }$ has 2 free dimensions, but a 2-class model has $2(2+1)-1=5$ parameters. Counting alone already shows a single 2-class family cannot be identified; the paper’s Table 1 gives explicit parameter values (Aleshin-Guendel 2020 - Latent Class Identifiability Theorem and LCMCR).

import itertools, numpy as np
def cell_probs(nu, lam):
    K = len(lam[0]); out = {}
    for h in itertools.product([0, 1], repeat=K):
        out[h] = sum(n*np.prod([l[k] if h[k] else 1-l[k] for k in range(K)]) for n, l in zip(nu, lam))
    return out
def conditional(P):
    p0 = P[(0,)*len(next(iter(P)))]; return p0, {h: v/(1-p0) for h, v in P.items() if any(h)}

Connections

• Link between the missing cell and the dark figure in the log-linear approach: Silverman 2020 - Poisson Log-Linear Model for Multiple Lists.
• Binomial-times-multinomial factorisation is the Fienberg (1972) conditional likelihood used in Concept - Closed-Population Capture-Recapture and Otis Models.

See Also

• Aleshin-Guendel 2020 - Mixed-Moments Theorem
• Aleshin-Guendel 2020 - Identifiability of Latent Class MSE Models - Overview