Dirichlet-Multinomial Mixture Regression

• Dirichlet-multinomial mixture regression (DMMR) is a family of models that combine Dirichlet-multinomial distributions with regression and latent mixture components to handle overdispersed count data.
• It employs techniques such as sparse regularization, zero-inflation handling, and nonparametric Bayesian methods to infer latent classes and improve identifiability.
• Empirical applications span microbiome analysis and discrete choice modeling, demonstrating robust performance in high-dimensional settings.

Searching arXiv for recent and foundational papers on Dirichlet-multinomial mixture regression and related models. Dirichlet-Multinomial Mixture Regression (DMMR) denotes a family of models that combine Dirichlet or Dirichlet-multinomial structure with regression and latent-mixture machinery. In one line of work, taxa counts are modeled by a Dirichlet-multinomial (DM) regression with sample-specific parameters linked to covariates through a log-linear form, primarily to account for overdispersion and to support variable selection in high-dimensional settings (Chen et al., 2013). In another, mixture regression is implemented nonparametrically through Dirichlet process priors over latent classes or random effects, with multinomial logit or categorical kernels and a data-driven number of occupied components [(Krueger et al., 2018); (Liverani et al., 2013)]. Closely related models condition Dirichlet priors on arbitrary metadata, as in feature-conditioned topic models (Mimno et al., 2012). Recent identifiability results show that unrestricted mixtures of Dirichlet and Dirichlet-multinomial distributions are not identifiable, and that fixed-total, box-type, or low-component restrictions are essential when unique interpretation of mixture parameters is required (Nguyen et al., 23 Mar 2026).

1. Conceptual scope and model families

The literature uses closely related constructions rather than a single universally fixed specification. One strand uses the Dirichlet-multinomial distribution as the sampling model for multivariate count data, with regression entering through covariate-dependent concentration parameters. A second strand uses a Dirichlet process mixture prior over latent classes in multinomial or categorical regression kernels. A third, related but distinct, strand replaces a fixed Dirichlet prior with a feature-conditioned Dirichlet prior, preserving the multinomial allocation structure while changing how document-specific mixing proportions are formed [(Chen et al., 2013); (Krueger et al., 2018); (Mimno et al., 2012)].

Formulation	Core mechanism	Representative source
Sparse DM regression	DM likelihood with log-linear covariate link and sparse group $\ell_1$ penalty	(Chen et al., 2013)
Zero-inflated DM regression	DM-based count model with latent structural-zero indicators and spike-and-slab priors	(Koslovsky, 2023)
Dirichlet process mixture multinomial logit	Truncated stick-breaking prior over latent classes in an MNL kernel	(Krueger et al., 2018)
Profile regression categorical special case	Dirichlet process mixture linking categorical response and discrete covariates through cluster membership	(Liverani et al., 2013)
Metadata-conditioned topic model	Log-linear regression for document-specific Dirichlet priors over topic proportions	(Mimno et al., 2012)

This suggests that DMMR is best understood as an umbrella label for regression models in which Dirichlet-multinomial structure interacts with latent mixture representations, covariate-conditioned concentration parameters, or both. The common ingredients are multinomial-type observations, simplex-constrained latent proportions, and a regression layer that alters either the Dirichlet parameters or the distribution over mixture components.

2. Dirichlet-multinomial regression as the parametric core

In sparse DM regression for microbiome data, observed taxa counts $\mathbf{Y}=(y_{ij})_{n \times q}$ are modeled using a DM distribution with sample-specific parameters $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$, where

$$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$

The associated log-likelihood, up to terms not involving the parameters, is

$$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$

The model is used to account for overdispersion of observed counts, and the DM regression model can be used for testing the association between taxa composition and covariates using the likelihood ratio test (Chen et al., 2013).

High-dimensional settings motivate penalized estimation. The sparse DM regression penalizes the negative log-likelihood by

$$pl(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}, \lambda_1, \lambda_2) = -l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) + \lambda_1 \sum_{k=1}^{p} \| \boldsymbol{\beta}_k \|_2 + \lambda_2 \sum_{k=1}^{p} \| \boldsymbol{\beta}_k \|_1,$$

where the group lasso term allows entire groups of covariates to be removed and the $\ell_1$ term induces sparsity within groups. The stated purpose is hierarchical selection: relevant covariates at the group level and the specific taxa associated with each retained covariate at the within-group level (Chen et al., 2013).

A major extension introduces excess-zero handling. The Bayesian zero-inflated DM regression model augments the standard DM representation with latent indicators $\eta_{ij} \in \{0,1\}$, setting $c_{ij}=0$ when $\eta_{ij}=0$ and $\mathbf{Y}=(y_{ij})_{n \times q}$0 when $\mathbf{Y}=(y_{ij})_{n \times q}$1. Regression enters through a log-link for abundance,

$\mathbf{Y}=(y_{ij})_{n \times q}$2

and a logistic regression for zero inflation,

$\mathbf{Y}=(y_{ij})_{n \times q}$3

The explicit distinction between structural and at-risk zeros addresses a limitation of the classical DM distribution, which is unable to handle excess zeros typically found in practice and may bias inference (Koslovsky, 2023).

3. Mixture regression and Bayesian nonparametric formulations

In Bayesian nonparametric mixture regression, a central construction places a Dirichlet process prior on the mixing distribution. The Dirichlet process mixture model of discrete choice specifies

$\mathbf{Y}=(y_{ij})_{n \times q}$4

$\mathbf{Y}=(y_{ij})_{n \times q}$5

so that heterogeneity is represented by a discrete mixing distribution whose complexity is inferred from the evidence rather than fixed a priori. The truncated stick-breaking representation induces shrinkage, and the vast majority of clusters are effectively empty unless strongly supported by the data (Krueger et al., 2018).

The corresponding marginal likelihood has the standard mixture form

$\mathbf{Y}=(y_{ij})_{n \times q}$6

with the multinomial logit kernel supplying the regression structure. The model is explicitly described as conceptually similar to DMMR in discrete choice models, where a Dirichlet process mixture prior is applied over the distribution of taste parameters in a multinomial logit kernel (Krueger et al., 2018).

A second nonparametric route is profile regression. In PReMiuM, the response vector and covariates are linked nonparametrically through cluster membership under a Dirichlet process mixture model,

$\mathbf{Y}=(y_{ij})_{n \times q}$7

and

$\mathbf{Y}=(y_{ij})_{n \times q}$8

For categorical outcomes with discrete covariates, the package states that DMMR is implemented by specifying yModel="Categorical" and xModel="Discrete" (Liverani et al., 2013).

Dependent Dirichlet processes via latent multinomial processes provide an additional foundation for regression-dependent mixtures. In that construction,

$\mathbf{Y}=(y_{ij})_{n \times q}$9

$$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$0

with dependence determined by neighborhood sets $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$1. The paper states that this method can be a foundation for DMMR because it supports mixture models with regression-dependent mixing distributions and controlled cross-index correlations (Nieto-Barajas, 2021).

4. Metadata-conditioned Dirichlet priors and topic-model analogues

A related model class arises in topic modeling. Dirichlet-multinomial regression (DMR) topic models retain the LDA framework but replace the fixed corpus-wide Dirichlet prior with a document-specific prior determined by observed features. For document $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$2 and topic $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$3,

$$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$4

where $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$5 may encode author, publication venue, references, dates, or other document features. Topic proportions are then drawn as

$$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$6

This is a fully conditional framework: it models the distribution of words given metadata, not the distribution of metadata itself (Mimno et al., 2012).

The generative process draws topic-word distributions $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$7 and regression weights $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$8, observes document features $$\boldsymbol{\gamma}(\mathbf{x}^i) = (\gamma_1(\mathbf{x}^i),\ldots,\gamma_q(\mathbf{x}^i))$$9, sets $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$0, draws $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$1, then samples topic assignments and words from multinomial distributions. After integrating out $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$2, the feature-dependent contribution to the likelihood is

$$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$3

The gradient with respect to $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$4 involves the digamma function and the sample-specific topic counts $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$5 (Mimno et al., 2012).

Although DMR is not identical to Dirichlet process mixture regression, it is structurally adjacent to DMMR because the Dirichlet parameters are regressed on observed features rather than held fixed. Its practical contribution is that new metadata types are incorporated by augmenting the feature vector $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$6, without requiring custom model construction or inference derivation for each new type of metadata. Empirically, the paper reports that DMR models with appropriately chosen features can meet or exceed the performance of several previously published topic models designed for specific data, including Author-Topic, Citation-Topic, and Topics-Over-Time, and that DMR outperforms vanilla LDA in all metrics when relevant metadata is present (Mimno et al., 2012).

5. Estimation, computation, and empirical behavior

The computational strategies used across DMMR-related models differ substantially. Sparse DM regression uses a block-coordinate descent algorithm built from a quadratic approximation of the log-likelihood, group-wise updates, closed-form thresholding, optional line search, and iteration until parameters stabilize. The stated objective is to solve a convex but non-differentiable penalized optimization problem efficiently in high dimensions (Chen et al., 2013).

Zero-inflated DM regression is fully Bayesian and uses Metropolis-Hastings within Gibbs. The update scheme includes latent Gamma variables, zero-inflation indicators, zero-inflation regression coefficients and inclusion indicators, DM regression coefficients and inclusion indicators, an Expand/Contract step for changing parameter-space dimension when $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$7 changes, and Polya-Gamma data augmentation for logistic regression updates. Covariates with marginal posterior probability of inclusion above threshold, for example $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$8, are called as truly associated (Koslovsky, 2023).

For Dirichlet process mixture multinomial logit, posterior inference is carried out by an expectation maximisation algorithm for MAP estimation. The E-step computes posterior responsibilities, while the M-step numerically updates the concentration parameter and optimizes a weighted multinomial logit objective for each component. The paper emphasizes that this avoids label switching and heavy simulation, and that it scales to larger datasets (Krueger et al., 2018).

PReMiuM uses blocked Gibbs sampling based on the slice sampler, the retrospective sampler, and their combination. It additionally implements Metropolis-Hastings label-switch moves, convergence diagnostics, prediction via pseudo-profiles, and variable selection for both discrete and continuous covariates (Liverani et al., 2013).

Empirical performance is heterogeneous across applications but consistently tied to the statistical role of the Dirichlet-multinomial structure. In microbiome analysis, sparse DM regression selected 11 nutrients and 13 genera as significantly associated in a dataset with gut microbiome sequencing from 98 individuals, 118 nutrient covariates, and counts for 30 common genera, and the predicted counts closely matched observed counts with $$\gamma_j(\mathbf{x}^i) = \exp\left(\sum_{k=0}^{p}\beta_{jk} x_{ik}\right).$$9 (Chen et al., 2013). In the zero-inflated setting, application to $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$0 samples, $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$1 microbial genera, and $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$2 dietary covariates with over $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$3 observed zeros captured associations for both abundance and zero-inflation probability (Koslovsky, 2023). In discrete choice, the Dirichlet process mixture multinomial logit model provided the best in-sample fit and cross-validated predictive performance in a case study on motorists’ route choice preferences (Krueger et al., 2018).

6. Identifiability, distinctions, and limitations

A central theoretical issue is identifiability. For Dirichlet densities on the interior of the simplex, the shift identity

$$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$4

implies that the mapping from mixing measure to mixture density is not injective on the full parameter space. The same shift identity propagates to the Dirichlet-multinomial kernel,

$$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$5

so unrestricted mixtures of Dirichlet-multinomial distributions are not identifiable (Nguyen et al., 23 Mar 2026).

The same source states that these implications extend directly to DMMR: if the model allows the Dirichlet parameters to vary freely, so that for each covariate value the parameter vector $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$6 ranges unrestrictedly over $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$7, then the mapping from parameters or regression functions to the implied data distribution is not injective. Identifiability is recovered on a fixed-total parameter slice $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$8, on restricted box-type regions in which each parameter varies in an interval of length less than $$l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) = \sum_{i=1}^n \Bigg[ \tilde{\Gamma}\left(\sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) - \tilde{\Gamma}\left(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j(\mathbf{x}^i; \boldsymbol{\beta}^j)\right) + \sum_{j=1}^q \left\{ \tilde{\Gamma}(y_{ij}+\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) - \tilde{\Gamma}(\gamma_j(\mathbf{x}^i;\boldsymbol{\beta}^j)) \right\} \Bigg].$$9, and for mixtures with fewer than $$pl(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}, \lambda_1, \lambda_2) = -l(\boldsymbol{\beta}; \mathbf{Y}, \mathbf{X}) + \lambda_1 \sum_{k=1}^{p} \| \boldsymbol{\beta}_k \|_2 + \lambda_2 \sum_{k=1}^{p} \| \boldsymbol{\beta}_k \|_1,$$0 atoms (Nguyen et al., 23 Mar 2026).

Several distinctions are important. Dirichlet regression for compositional outcomes models vectors of positive proportions that sum to one and typically uses a multinomial logit link for the mean, whereas standard multinomial models do not capture the continuous proportional outcome and Dirichlet-multinomial regression is oriented toward overdispersed counts or toward a prior for counts (Saddiki et al., 27 Mar 2025). Classical DM regression also cannot truly model structural zeros; the zero-inflated DM formulation addresses this by separating structural from at-risk zeros (Koslovsky, 2023).

A common misconception is that any regression involving a Dirichlet prior and multinomial observations is interchangeable with DMMR. The available literature suggests a narrower reading. Feature-conditioned topic models, sparse DM regression for counts, zero-inflated DM regression, and Dirichlet process mixture multinomial logit models all share Dirichlet-multinomial structure, but they target different inferential objects: document-topic priors, overdispersed taxa counts, structural zero processes, or latent preference classes. Their overlap is substantive, but their likelihoods, priors, and identifiability regimes are not identical [(Mimno et al., 2012); (Chen et al., 2013); (Krueger et al., 2018); (Nguyen et al., 23 Mar 2026)].