Deep Generalised Mixed Models: a Novel Neural Network Structure for Analysing Hierarchical Data

Dimitris Rizopoulos; Loes Keijsers; Manon Hillegers; Nina van Gerwen; Sten Willemsen

REVIEW 3 major objections 6 minor 17 references

Deep Generalised Mixed Models: a Novel Neural Network Structure for Analysing Hierarchical Data

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The Deep Generalised Mixed Model estimates both fixed and random effects with neural networks and imputes missing responses from the model's posterior predictive distribution, keeping inference valid under missing-at-random dropout.

desk verdict A useful neural extension of mixed models with a data-augmentation step for MAR dropout, but the headline claim of valid inference under MAR is asserted rather than demonstrated. read the letter →

arxiv 2608.05930 v2 pith:Y3PUEHG3 submitted 2026-08-06 stat.ML cs.LG

Nina van Gerwen , Dimitris Rizopoulos , Manon Hillegers , Loes Keijsers , Sten Willemsen This is my paper · ORCID

classification stat.MLcs.LG

keywords mixedeffectsmodelsneuralnetworkslongitudinaldatamissing-at-randomvariationalauto-encodersexperiencesamplingmethodaugmentationdynamicpredictions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes the Deep Generalised Mixed Model (DGMM), a neural-network architecture that extends mixed effects models to deep learning by replacing both the fixed-effects and random-effects parts with feed-forward neural networks. The central claim is that this architecture can analyse intensive longitudinal data from generic outcome distributions, scale to high-dimensional settings, and give valid inference and dynamic predictions when dropout is missing at random (MAR). Estimation uses a variational auto-encoder: an encoder maps each subject's observed data to a variational posterior over random effects, and a decoder turns draws from that posterior, together with time, into subject-specific mean shifts. Missing responses are imputed during training by sampling from the model's posterior predictive distribution, following a stochastic EM-style data augmentation algorithm. On simulations under MAR the method can recover true trajectories with less dynamic bias than a recurrent neural network when it converges, but the paper reports substantial run-to-run instability, posterior collapse, and a failure to forecast individual states well on the motivating GrowIt! data.

What carries the argument

The central object is the three-network DGMM: a fixed-effects network $h$, an encoder network $e$, and a decoder network $d$. The encoder outputs the means and variances of a normal variational posterior for the random effects, the decoder maps a draw $z$ (held constant for a subject) together with measurement time into the subject-specific part $\mu_2$, and the fixed plus random parts are summed and passed through link functions to form predictions for each outcome. Estimation maximizes the ELBO, with the KL term available in closed form because both the variational posterior and the prior are normal. For missing data, the load-bearing mechanism is the data augmentation algorithm in Section 2.3: each epoch imputes the missing responses by sampling from $p(Y_i^m \mid Y_i^o; \theta)$, implemented with an adaptive Metropolis-Hastings step, and then retrains on the completed input.

What would settle it

Simulate many datasets from a known linear mixed model with a documented MAR dropout rule, fit the DGMM with several random seeds and large sample sizes, and check whether the estimated fixed-effect curves and random-effect posteriors concentrate around the true values as subjects and within-subject measurements grow; if repeated runs remain scattered or imputation error does not approach the true model's imputation error, the valid-inference-under-MAR claim is refuted.

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Extended reading notes

Core claim

The paper's claim is that the DGMM—a network in which $\mu_1(X_i, T_i)$ is a neural-net fixed-effects function and $\mu_2(b_i, T_i)$ is a neural-net random-effects function, with random effects $b_i \sim N(0,I)$—can serve as a generalised mixed model for longitudinal outcomes with arbitrary response distributions. Because the marginal likelihood is intractable, the model maximizes an evidence lower bound: Monte-Carlo approximation of the expected log-likelihood minus the closed-form KL divergence between the variational posterior $N(\mu_b(H_i;\phi), \Sigma_b(H_i;\phi))$ and the standard normal prior. Missing-at-random dropout is handled by a data augmentation loop that alternates between optimizing this bound on the currently imputed data and drawing new imputations of the missing responses from the posterior predictive distribution using a random-walk Metropolis-Hastings sampler. The paper asserts that this gives semi-parametric modelling of the mean and correlation structure, scalability to high-dimensional ESM data, and valid inference under MAR missingness.

Load-bearing premise

The load-bearing premise is that imputing missing responses from the model's own posterior predictive distribution, together with a normal approximation to the random-effects posterior, forms a valid EM-type estimation procedure under missing-at-random data, and the paper provides no convergence proof for this loop.

Editorial extensions

If this is right

A single DGMM architecture can model longitudinal outcomes that follow different distributions (Gaussian, Student-$t$, gamma, beta, binomial, Poisson, negative binomial), removing the need to hand-specify a separate model for each outcome.
The encoder pools all available measurements per subject, so the model can generate subject-specific dynamic predictions past the last observation while the data augmentation is designed to keep those predictions unbiased under MAR dropout.
When training converges, the simulations indicate the DGMM recovers true trajectories with lower dynamic bias than an RNN under MAR mechanisms and can approach the predictive accuracy of the true data-generating mixed model.
The architecture is not yet stable enough for routine use: simulation runs show large variance with occasional under- or overfitting, posterior collapse is a known failure mode, and on the GrowIt! data neither the DGMM nor the RNN forecasted individual negative states well.
Because the decoder is a generic network, future users can replace its feed-forward layers with recurrent or convolutional layers, extending the same formulation to other forms of nested or hierarchical data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The normal variational posterior for the random effects is probably the weakest statistical link; replacing it with a heavier-tailed or more flexible variational family (for example a Student-$t$ or a normalizing-flow posterior) could reduce bias for the strongly skewed ESM outcomes.
The large run-to-run variance suggests the stochastic EM-style update lacks a convergence diagnostic; pooling imputations across multiple seeds or adding a proper convergence check on the imputed-data bound could make the method usable before further architectural changes.
The encoder's latent means can be read as low-dimensional embeddings of individual trajectories, so the same architecture could be extended to joint modelling of longitudinal and time-to-event data or to clustering patients by trajectory shape.
A natural testable extension is to benchmark the DGMM against standard missing-data tools such as multiple imputation by chained equations or joint mixed models on the same MAR data, since those are established comparators the paper does not include.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, it does something genuinely new: it extends Mandel et al.'s neural-network fixed effects to random effects via a VAE, and adds a data-augmentation step aimed at MAR missingness. Second, the central claim in the abstract—"provide valid inference when data are missing-at-random"—is not supported by the theory or the simulations. The authors are honest about instability, but the claim overreaches.

What the paper does well: the model formulation is clear and the ELBO derivation is standard and correct. The simulation setup is sensible, and the application to the GrowIt! ESM study is real, with a thoughtful discussion of why ESM data may not suit prediction. The authors openly list limitations, including posterior collapse and hyperparameter sensitivity. That transparency is a credit.

The soft spots are real but fixable. First, Algorithm 1 is a stochastic EM where the M-step is itself a variational approximation; there is no convergence theorem, so the fixed point need not be the observed-data MLE. Second, the normal approximation to p(b_i | Y_i) is only exact for linear Gaussian models, yet it is used generally. Third, the simulations evaluate RMSPE and point-prediction bias, not parameter bias, interval coverage, or hypothesis-test behavior. So the phrase "valid inference" is never actually tested. There is also a typo in Eq. (2): the integrand repeats the left-hand side; it should be p(Y^m_i | b_i; theta). That is minor, but it makes you wonder if the missing-data derivation was checked carefully. Finally, no code is released; the data are restricted, but code could be shared. The run-to-run variance in Figure 4 is worrying; the authors concede it, but it means the method is not yet reliable.

Who this is for: people working on ESM/mHealth missing data and on neural extensions of mixed models. I would send it to a serious referee, because the idea is promising and the problems are addressable. My recommendation is to accept it for peer review with an expectation of major revision: either supply a convergence guarantee or sharply weaken the claim, and add simulations that report parameter coverage and interval properties.

Referee Report

3 major / 6 minor

Summary. The paper proposes the Deep Generalised Mixed Model (DGMM), a variational-autoencoder-based neural network that generalizes mixed effects models by representing both fixed and random effects with neural networks. For missing longitudinal outcomes, the authors introduce a stochastic data-augmentation algorithm (Algorithm 1) that imputes missing values from the model's posterior predictive distribution, motivated by missing-at-random (MAR) drop-out in the GrowIt! experience sampling study. The method is applied to the GrowIt! data and evaluated in simulations against an LSTM RNN and the true linear mixed model, using RMSPE and dynamic bias. The paper reports that the DGMM shows potential but suffers from model instability and suboptimal performance in the motivating application.

Significance. If the advertised property of valid inference under MAR were rigorously established, the DGMM would be a meaningful contribution at the intersection of mixed models and deep learning, extending Mandel et al. (2021) by allowing both fixed and random effect parts to be nonlinear and by providing a principled missing-data mechanism. The paper is commendably honest about its limitations, explicitly reporting instability, hyperparameter sensitivity, and underperformance in the GrowIt! analysis. However, the significance is currently limited by (i) the absence of theoretical support for the central MAR inference claim, (ii) an evaluation that measures only predictive accuracy rather than inferential properties such as coverage, and (iii) the lack of uncertainty quantification in the reported comparisons. The core ideas are promising, but the evidence presented does not yet justify the abstract's strongest claims.

major comments (3)

[Section 2.3, Eq. (2)] Equation (2) is internally inconsistent as printed: it states p(Y^m,_i^{(v)} | Y^o_i; θ^{(v)}) = ∫ p(Y^m,_i^{(v)} | Y^o_i; θ^{(v)}) p(b_i | Y^o_i; θ^{(v)}) db_i, meaning the left-hand side appears as a factor inside the integrand. The intended expression is presumably p(Y^m_i | Y^o_i; θ) = ∫ p(Y^m_i | b_i; θ) p(b_i | Y^o_i; θ) db_i, using the conditional-independence assumption of Eq. (1). Since Eq. (2) defines the imputation distribution used in Step 2 of Algorithm 1—the core of the MAR claim—this must be corrected and reconciled with the sampling algorithm in Web Appendix C.
[Abstract and Section 5] The Abstract's claim that the DGMM 'provides valid inference when data are missing-at-random' is not established by the paper. Algorithm 1 is a stochastic EM-type loop, but its Step 1 performs a variational (approximate) M-step, and no convergence result is supplied to show that the fixed point of the loop maximizes the observed-data ELBO or marginal likelihood under MAR. In addition, the normal variational family for p(b_i | Y_i; θ) is only exact for Gaussian linear models; for generic outcome distributions F_k in Eq. (1) the approximation error is not quantified. The paper should either provide a theorem or a precise convergence argument, or the claim should be softened to predictive performance rather than valid frequentist inference.
[Section 4] The simulation studies evaluate RMSPE and dynamic bias for point predictions (Eq. (4)) but never assess inferential properties such as parameter bias, interval coverage, or confidence-interval width for the fixed effects or random-effect covariance. Thus the headline claim of valid inference under MAR is never empirically tested. Moreover, Figure 4 shows large run-to-run variance, and the line graphs in Figure 5 are presented without confidence bands or standard errors, making it difficult to judge whether DGMM–RNN differences are meaningful. I recommend adding uncertainty measures for all reported metrics and, if the 'valid inference' claim is retained, including inferential evaluation such as coverage of the true trajectory or of model parameters.

minor comments (6)

[Section 2.2] The closed-form KL divergence sums over k with terms μ_il and σ^2_il, but the latent dimension is u; please use a consistent index (e.g., l = 1, ..., u) and avoid reusing k, which already denotes the outcome index.
[Section 2.4] In the dynamic prediction formula, the inner integral is over p(b_j | Y_j(t); θ) but written with db_i, and the outer integral over p(θ | D_n) uses dθ; please make the integration variables consistent (db_j and dθ) for readability.
[Section 3] The application uses a simplified architecture that omits the fixed-effects network h and adds baseline covariates to the decoder, which differs from the general model in Section 2 and Figure 1. Please explain this simplification and its implications for identifiability of fixed versus random effects in the reported results.
[Figures 2, 3, and 5] Several figure labels and captions contain garbled text (e.g., 'Sim)la(on S()dy I Res)l(s', 'T me ( u )'), which appears to be a rendering artifact; these need to be repaired before publication.
[Web Appendices] Key implementation details, such as the inner steps of the Random Walk Metropolis–Hastings algorithm and the Robbins–Monro process in Step 2 of Algorithm 1, are relegated to Web Appendix C, which is not included in the manuscript version under review; the main text should summarize these steps or the appendices should be available for verification.
[References] Reference 34 is cited as 'Adam, K. D. B. J. et al.'; the correct citation is Kingma, D. P. and Ba, J., 'Adam: A method for stochastic optimization,' which should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the DGMM derivation is self-contained, with the unsupported MAR-inference claim being a correctness gap rather than a circular step.

full rationale

The paper's derivation chain is self-contained against its own inputs. The model is defined in Eq. (1) as a generalised mixed model with NN fixed and random effect functions; the ELBO in Section 2.2 is derived from the marginal likelihood by standard variational arguments; the normal variational family is stated as an approximation with its exactness condition given; and the missing-data routine in Algorithm 1 is a stochastic-EM-style self-consistency loop that imputes missing values from the model's own posterior predictive distribution. This is not circular: the imputed values are auxiliary training inputs, not the quantities being predicted or evaluated. Predictions are assessed on independent test sets via RMSPE and bias, so no fitted parameter is renamed as a prediction. No uniqueness theorem is imported from the authors' prior work, and the cited prior work by Mandel et al. is external to the authors. The equality of the ELBO with the marginal log-likelihood when q(b_i) equals the true posterior is a mathematical identity, not a reduction of the result to its inputs. Equation (2) as printed repeats the left-hand side inside the integrand, but this is a typographical error in the stated factorisation rather than a circular definition; the intended conditional independence factor p(Y^m_i | b_i; theta) p(b_i | Y^o_i; theta) is standard. The abstract's claim of 'valid inference when data are missing-at-random' is under-supported, because Algorithm 1 lacks a convergence proof and the simulations evaluate point predictions rather than interval coverage or estimator bias, but an unsupported claim is a correctness risk, not circularity. Accordingly, no specific circular step can be quoted and exhibited, and the appropriate score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The DGMM relies on standard variational inference machinery plus several domain assumptions about the missingness mechanism and the distribution of random effects. Hyperparameters are hand-chosen and not tuned systematically, contributing to instability.

free parameters (4)

Latent dimension u = 5 (GrowIt! analysis)
Dimension of the random effects vector; hand-chosen. Larger u increases model capacity but also KL complexity; no systematic tuning is reported.
Encoder architecture = 4 hidden layers, 64 units each
Depth and width chosen by hand for the encoder network; affects quality of the variational posterior approximation.
Decoder architecture = 2 hidden layers, 48 units each
Chosen shallow deliberately to combat posterior collapse; a modeling decision that trades flexibility for stability.
KL annealing schedule = Cyclic KL-annealing with beta-VAE weighting; warm-up before imputation
Hand-selected to aid convergence and prevent posterior collapse; the schedule is not derived from data and influences results.

assumptions (5)

domain assumption Outcomes are conditionally independent given random effects bi
Core conditional independence assumption in Eq. (1) that defines the mixed model structure.
domain assumption Random effects follow N(0, I)
Prior imposed in Section 2; differs from standard mixed models where bi ~ N(0, D), and restricts the covariance structure.
domain assumption Missingness is missing-at-random (MAR) and the set of time points is pre-specified
Assumed in Section 2.3 to justify the data augmentation algorithm; if missingness depends on unobserved values (MNAR), the method may be biased.
standard math Posterior p(bi | Yi; theta) can be approximated by a multivariate normal (Bayesian CLT)
Used in Section 2.2 to justify the normal variational distribution; exact only for linear Gaussian models.
standard math ELBO maximization is equivalent to maximizing the marginal log-likelihood
Standard variational inference result invoked in Section 2.2 to derive the training objective.

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Cite this review

Pith. "Pith review of Deep Generalised Mixed Models: a Novel Neural Network Structure for Analysing Hierarchical Data." pith.science (2026). https://pith.science/paper/Y3PUEHG3

@misc{pith2026260805930,
  author       = {Pith},
  title        = {Pith review of: Deep Generalised Mixed Models: a Novel Neural Network Structure for Analysing Hierarchical Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3PUEHG3}},
  note         = {Machine review of arXiv:2608.05930}
}

read the original abstract

The experience sampling method (ESM) is a longitudinal research design where participants report their thoughts, emotional states and behaviours multiple times a day. Our work is motivated by such data collected by the GrowIt! app, which was released to investigate daily emotions among adolescents during the COVID-19 pandemic. Current procedures to analyse ESM data face various challenges. While standard statistical techniques may not scale well to a high-dimensional setting, machine learning procedures can give biased results due to selection bias introduced by missingness. In our motivating dataset, adolescents dropped out due to previous strong feelings of negative emotions. Hence, the implied missing data are of the missing-at-random type that standard machine learning procedures cannot accommodate. We develop a novel neural network architecture that generalises mixed effects models to deep learning to overcome these challenges. It allows semi-parametric and flexible modelling of data's mean and correlation structure through fixed and random effects. For estimation, we use an adaptation of variational auto-encoders and a Bayesian data augmentation algorithm. Through this approach, the model can accommodate longitudinal outcomes following generic distributions, scale well to high-dimensional settings and provide valid inference when data are missing-at-random. We applied the Deep Generalised Mixed Model to the GrowIt! study and various simulations. The results show potential for the Deep Generalised Mixed Model, yet suboptimal performance due to model instability.

Figures

Figures reproduced from arXiv: 2608.05930 by the authors.

**Figure 2.** Individualised dynamic predictions for three outcomes for three individuals in the GrowIt! test set at [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Plot (a) on the left depicts the RMSE in the train data for the three tested methods. Plots (b) on [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Boxplots of the RMSE in the training data in Simulation Study II for the three methods under [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Line graphs of the average RMSPE(u, t) and bias(u, t) for t = {5, 15} in an independent test set for the three methods under a proportion of 0.30 missing data and a probabilistic drop-out data generating mechanisms. The line graphs in the left column are estimated over…

**Figure 6.** Figure 6: RMSE of the data augmentation algorithm in Simulation Study II for the DGMM under varying [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

Review history (2 revisions) →

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

The Grow It! app— longitudinal changes in adolescent well-being during the COVID-19 pandemic: a proof-of-concept study

(1) Dietvorst, E; Legerstee, J.; Vreeker, A; Koval, S; Mens, M.; Keijsers, L; Hillegers, M. The Grow It! app— longitudinal changes in adolescent well-being during the COVID-19 pandemic: a proof-of-concept study. European Child & Adolescent Psychiatry 2023, 32, 1097–1107. 18 (2) Dietvorst, E.; Hillegers, M. H.; Legerstee, J. S.; Vries, L. P. D.; Vreeker, A...

2023

[2] [4]

New developments in experience sampling methodology

(11) Tuerlinckx, F.; Kuppens, P.; Ariens, S.; Cloos, L.; Dejonckheere, E.; Lafit, G.; Niemeijer, K.; Revol, J.; Schat, E.; Schreuder, M., et al. New developments in experience sampling methodology. British Journal of Mathematical and Statistical Psychology 2026, 79, 46–65. (12) Prince, S. J., Understanding deep learning ; MIT press:

2026

[3] [8]

A solution for missing data in recurrent neural networks with an application to blood glucose prediction

(17) Tresp, V.; Briegel, T. A solution for missing data in recurrent neural networks with an application to blood glucose prediction. Advances in Neural Information Processing Systems 1997,

1997

[4] [10]

Speech recognition with missing data using recurrent neural nets

(18) Parveen, S.; Green, P. Speech recognition with missing data using recurrent neural nets. Advances in Neural Information Processing Systems 2001,

2001

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C.; Kale, D.; Wetzel, R

(19) Lipton, Z. C.; Kale, D.; Wetzel, R. In Machine learning for healthcare conference , 2016, pp 253–270. (20) Choi, E.; Bahadori, M. T.; Schuetz, A.; Stewart, W. F.; Sun, J. In Machine learning for healthcare conference, 2016, pp 301–318. (21) Le Morvan, M.; Josse, J.; Scornet, E.; Varoquaux, G. What’s a good imputation to predict with missing values? A...

2016

[6] [80]

Recurrent neural networks for missing or asynchronous data

(16) Bengio, Y.; Gingras, F. Recurrent neural networks for missing or asynchronous data. Advances in neural information processing systems 1995,

1995

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Cyclical annealing schedule: A simple approach to mitigating kl vanishing

(35) Fu, H.; Li, C.; Liu, X.; Gao, J.; Celikyilmaz, A.; Carin, L. Cyclical annealing schedule: A simple approach to mitigating kl vanishing. arXiv preprint arXiv:1903.10145

arXiv 1903

[8] [1979]

A new class of stochastic EM algorithms

(32) Allassonnière, S.; Chevallier, J. A new class of stochastic EM algorithms. Escaping local maxima and handling intractable sampling. Computational Statistics & Data Analysis 2021, 159, 107159. (33) Garthwaite, P. H.; Fan, Y.; Sisson, S. A. Adaptive optimal scaling of Metropolis–Hastings algorithms using the Robbins–Monro process. Communications in Sta...

2021