Aggregated Posterior Predictive Checks for Generative Modeling

Shweta Dutta; Gemma E. Moran

stat.MESep 17, 2026

Aggregated Posterior Predictive Checks for Generative Modeling

Authors: Shweta Dutta, Gemma E. Moran

Organizations: Department of Statistics, Rutgers University

Abstract

Latent variable generative models are commonly fit using simple priors over latent variables, but draws from these priors often fail to produce realistic data. This failure is due to a mismatch between the prior and the aggregated posterior, the distribution of latent variables induced by the fitted model and the data. This mismatch is often viewed as evidence that the prior is misspecified and should be replaced. Alternatively, in modern generative models, a two-stage strategy is increasingly used where first, the model is fit, and second, the aggregated posterior is estimated (van den Oord et al.,2017; Rombach et al., 2022.). Synthetic data are then obtained by sampling from this aggregated posterior instead of the prior. To check such procedures, we introduce the aggregated posterior predictive check (APPC). Theoretically, we establish sufficient conditions under which the APPC is asymptotically calibrated. For probabilistic principal component analysis, we show that the APPC can remain calibrated under a misspecified latent prior when pervasive factors permit recovery of the signal space. Experiments with variational autoencoders show that aggregated posterior sampling improves generation for heavy-tailed and clustered data relative to Gaussian prior sampling while performing comparably to models with more flexible latent priors.

Figures & tables

\displaystyle\widehat{p}_{\mathrm{APPC}}=\frac{1}{R}\sum_{r=1}^{R}\mathbf{1}\left[d(\bm{X}^{(r),\mathrm{rep}})\geq d(\bm{X}^{\mathrm{new}})\right];

Algorithm 1 Aggregated posterior predictive check

Panel of Figure 1 — Figure 1: PPCA example. The empirical aggregated posterior predictive generates realistic samples in both well-specified and misspecified cases. The appc $p$ -values are approximately uniform in both cases, unlike prior or conditional-replicate sampling. Top row: example draws of latent variables $\bm{z}$ (up to an invertible linear transformation). Bottom row: $p$ -values from predictive checks over 500 trials. Kolmogorov-Smirnov tests of uniformity are in red.

Reference distribution $Q_{m}$	Reference mean $\bm{\mu}_{m}^{\mathrm{rep}}$	Reference covariance $\bm{V}_{m}^{\mathrm{rep}}$
$P_{0}K_{\theta^{\star},\mathrm{post}}$	$\bm{B}^{\star}\bm{\mu}_{0}$	$\bm{V}_{m}^{\mathrm{oracle}}=\bm{R}^{\star}+\bm{B}^{\star}\bm{\Sigma}_{0}\bm{B}^{\star}$
$P_{0}K_{\widehat{\theta}_{m},\mathrm{post}}$	$\widehat{\bm{B}}_{m}\bm{\mu}_{0}$	$\bm{V}_{m}^{\mathrm{learn}}=\widehat{\bm{R}}_{m}+\widehat{\bm{B}}_{m}\bm{\Sigma}_{0}\widehat{\bm{B}}_{m}$
$\widehat{P}_{\mathrm{agg},m}K_{\widehat{\theta}_{m},\mathrm{post}}$	$\widehat{\bm{B}}_{m}\overline{\bm{x}}_{m}^{\mathrm{agg}}$	$\bm{V}_{m}^{\mathrm{emp}}=\widehat{\bm{R}}_{m}+\widehat{\bm{B}}_{m}\widehat{\bm{\Sigma}}_{\mathrm{agg},m}\widehat{\bm{B}}_{m}$

Table 1: PPCA reference distributions.

Figure 2: Bivariate t. Top: Draws of synthetic data $\bm{X}^{\mathrm{rep}}$ . Bottom: $p$ -values from model checks over $200$ trials. G-EAP and G-DAP $p$ -values are not uniform but have similar performance as more complex generative models. Here, $N_{\mathrm{train}}=N_{\mathrm{agg}}=10000,N_{\mathrm{new}}=2000$ .

Figure 3: Clustered t. Top: Draws of synthetic data $\bm{X}^{\mathrm{rep}}$ . Bottom: $p$ -values from model checks over $200$ trials. G-EAP and G-DAP $p$ -values are not uniform but show similar behavior to more complex generative models. Here, $N_{\mathrm{train}}=N_{\mathrm{agg}}=10000,N_{\mathrm{new}}=2000$ .

Figure 4: Financial. Top: Aggregated posterior with standard VAE (learnable variance, 1000 epochs) produces synthetic $\bm{X}^{\mathrm{rep}}$ which visually capture the heavy-tailed behavior of financial data. Bottom: G-EAP $p$ -values are approximately uniform. Here, $N_{\mathrm{train}}=N_{\mathrm{agg}}=4000,N_{\mathrm{new}}=2000$ .

Figure 5: MNIST. Top: G-EAP produces recognizable digits. Bottom: G-EAP $p$ -values are approximately uniform. Here, $N_{\mathrm{train}}=N_{\mathrm{agg}}=10000,N_{\mathrm{new}}=2000$ .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: PPCA example (fourth cumulant diagnostic). The appc is calibrated; the empirical aggregated posterior sampling strategy generates realistic samples in both well-specified and misspecified cases. Top row: example draws of latent variables $\bm{z}$ . Bottom row: $p$ -values from predictive checks over 500 trials.

Figure 7: GMM. Top: Draws of data $\bm{x}$ . Aggregated posterior with standard VAE (learnable variance, 700 epochs) performs better compared to VAEs with true or mixture-based priors. Bottom: $p$ -values from model checks over $200$ trials. Here, $N_{\mathrm{train}}=N_{\mathrm{agg}}=10000,N_{\mathrm{new}}=2000$ .

Hyperparameter	Bivariate $t_{2}$	Bivariate clustered $t_{2}$	Bivariate Gaussian mixture	Financial	MNIST
Input Dimension	2	2	2	2	784
Latent Dimension	2	2	2	64	64
Hidden Dimension	128	128	128	128	400
Epochs	700	700	700	1000	350
Batch Size	256	256	256	64	256
Learning Rate	0.001	0.001	0.001	0.0001	0.0001

Appendix

Table 2: Hyperparameter settings of VAE for the datasets

Hyperparameter	Bivariate $t_{2}$	Bivariate clustered $t_{2}$	Bivariate Gaussian mixture	Financial	MNIST
T	100	100	100	100	100
Hidden Dimension	128	128	128	128	256
Epochs	700	500	700	300	200
Batch Size	256	256	256	128	256
Learning Rate	0.001	0.001	0.001	0.0001	0.0001

Appendix

Table 3: Hyperparameter settings of LDM for the datasets

Hyperparameter	stVAE	t3VAE	Clustered tVAE	VAE with GMM prior	VampVAE
Degrees of freedom for $t$	3	3	x	x	x
MC KL samples	5	x	x	x	x
Number of components	x	x	5	5	x
Number of Pseduo Inputs	x	x	x	x	64

Appendix

Table 4: Hyperparameter settings of models.

Experimental Setting	Bivariate $t_{2}$	Bivariate clustered $t_{2}$	Bivariate Gaussian mixture	Financial	MNIST
Training samples ( $N_{\mathrm{train}}$ )	10000	10000	10000	4000	10000
Aggregated samples ( $N_{\mathrm{agg}}$ )	10000	10000	10000	4000	10000
Replicate samples ( $N_{\mathrm{new}}$ )	2000	2000	2000	2000	2000
Number of trials ( $N_{\mathrm{trial}}$ )	200	200	200	200	200
Number of repetitions ( $B$ )	100	100	100	100	100
MC LDM samples	20	20	20	20	20

Appendix

Table 5: Experimental settings for the datasets

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Model Misspecification Variational Inference

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Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

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