MissBGM: Bayesian Generative Imputation

• MissBGM is a Bayesian generative imputation framework that jointly models both the data-generating process and the missingness mechanism to robustly handle MNAR scenarios.
• It utilizes an alternating optimization strategy with gradient-based latent variable updates and HMC sampling to achieve uncertainty-aware imputations.
• Empirical evaluations on synthetic and real-world tabular datasets demonstrate superior RMSE performance and improved uncertainty calibration compared to traditional methods.

MissBGM is a missing-data imputation framework built on Bayesian generative modeling. It is designed to fill in missing entries in a probabilistically coherent and uncertainty-aware manner, including settings in which missingness may be non-ignorable (MNAR). Its defining feature is explicit joint modeling of both the data-generating process and the missingness mechanism, so that the posterior over missing entries is informed by the observed values and by the mask itself rather than by observed data alone (Liu, 3 May 2026).

1. Problem setting and motivation

MissBGM addresses the standard distinction among MCAR, MAR, and MNAR missingness regimes. Under ignorable missingness assumptions such as MCAR and MAR, it is valid to infer missing values from observed data alone:

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) = p(X_{\text{mis}} \mid X_{\text{obs}}).$$

Under MNAR,

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$

so the mask $R$ itself contains information about the missing values. Classical imputers and many deep generative imputers either ignore this or treat the mask only as an input feature, which can yield biased imputations and poor uncertainty quantification (Liu, 3 May 2026).

The framework is therefore motivated by two requirements. First, it models the data-generating process. Second, it models the missingness mechanism. In the formulation used for MissBGM, these two components are coupled in a single latent-variable Bayesian model. This makes the method suitable for complex tabular data and for structured or non-ignorable missingness. A plausible implication is that MissBGM is aimed not merely at accurate point reconstruction, but at posterior inference for incomplete data.

2. Joint Bayesian generative model

For each sample $i$, MissBGM introduces a latent variable $z_i \in \mathbb{R}^d$, a complete data vector $x_i \in \mathbb{R}^p$, and a mask $r_i \in \{0,1\}^p$. The joint model is

$$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$

with

$\pi(z_i)=\mathcal N(0,I_d).$

Here $p_\theta(x_i\mid z_i)$ generates the data and $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$0 generates the missingness pattern from the full data. The posterior over missing entries is consequently informed by both the observed values and the mask (Liu, 3 May 2026).

For continuous data, the paper uses a Gaussian decoder:

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$1

typically with diagonal covariance

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$2

A neural network $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$3 outputs the mean and variance, with one head for $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$4 and one head for $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$5. For discrete variables, Bernoulli or Categorical likelihoods can be used.

The missingness mask is modeled explicitly by another neural network $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$6:

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$7

where

$$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$8

Because $$p(X_{\text{mis}} \mid X_{\text{obs}}, R) \neq p(X_{\text{mis}} \mid X_{\text{obs}}),$$9 is allowed to depend on the full completed data vector $R$0, MNAR patterns can be represented directly.

The framework also supports Bayesian neural networks for $R$1 and $R$2, placing priors on network weights and using variational inference. The reported experiments indicate that the deterministic version performed slightly better or similarly.

3. Alternating optimization and posterior sampling

The posterior target relevant to imputation is

$R$3

but MissBGM works with the full joint posterior

$R$4

which is not tractable in closed form. The method therefore uses an alternating optimization and posterior-sampling strategy (Liu, 3 May 2026).

The training scheme iteratively updates latent variables $R$5, missing values $R$6, and model parameters $R$7. For latent variables, the conditional log posterior is

$R$8

and under a diagonal Gaussian data likelihood the objective becomes

$R$9

These latent-variable updates are performed by gradient descent, sample-wise and in parallel.

For missing values, the conditional log posterior is

$i$0

with objective

$i$1

The first term fits the data model, the second makes the imputation consistent with the missingness mechanism, and $i$2 is a temperature or weight controlling how strongly the mask influences imputation. Observed coordinates are kept fixed; only missing entries are updated.

For model parameters, if Bayesian neural networks are used, MissBGM optimizes weighted ELBOs:

$i$3

$i$4

with Gaussian variational families

$i$5

If Bayesian neural networks are not used, the updates reduce to standard stochastic gradient ascent.

The implementation uses an EGM warm start: missing entries are initialized with KNN, an encoder-generator pair is trained to stabilize latent space and generator, and then $i$6 initializes the latent variables. Optimization is done with Adam. Reported implementation details include gradient clipping for sample-specific steps, batch normalization and LeakyReLU in the generator, and a softplus variance head with a floor for positivity. The default hyperparameters reported are latent dim $i$7, epochs $i$8, batch size $i$9, $z_i \in \mathbb{R}^d$0, generator hidden units $z_i \in \mathbb{R}^d$1, missingness network hidden units $z_i \in \mathbb{R}^d$2, learning rates $z_i \in \mathbb{R}^d$3 for $z_i \in \mathbb{R}^d$4 and $z_i \in \mathbb{R}^d$5 for $z_i \in \mathbb{R}^d$6, inner steps per batch $z_i \in \mathbb{R}^d$7, KL weight for BNNs $z_i \in \mathbb{R}^d$8, and MCMC settings of 1000 burn-in plus 1000 retained samples, step size $z_i \in \mathbb{R}^d$9, and 5 leapfrog steps.

After training, MissBGM performs sample-specific posterior inference on incomplete test cases. It first computes a MAP-like solution by iterating only the latent and missing-value updates, then samples from the posterior using Hamiltonian Monte Carlo within Gibbs: update $x_i \in \mathbb{R}^p$0 with HMC conditional on current $x_i \in \mathbb{R}^p$1, and update $x_i \in \mathbb{R}^p$2 with HMC conditional on updated $x_i \in \mathbb{R}^p$3. Posterior summaries are then computed from samples $x_i \in \mathbb{R}^p$4, including the posterior mean

$x_i \in \mathbb{R}^p$5

and the prediction interval

$x_i \in \mathbb{R}^p$6

4. Tempered objective and consistency result

MissBGM includes a theoretical analysis for the alternating procedure based on a tempered profiled objective. For an observed pair $x_i \in \mathbb{R}^p$7, the paper defines

$x_i \in \mathbb{R}^p$8

The empirical and population objectives are

$x_i \in \mathbb{R}^p$9

and

$r_i \in \{0,1\}^p$0

The consistency theorem is stated for this tempered target rather than for arbitrary misspecification and arbitrary $r_i \in \{0,1\}^p$1 (Liu, 3 May 2026).

Informally, under regularity assumptions including i.i.d. sampling, continuity, uniform convergence, sieve approximation, vanishing optimization error, and a separation or identifiability condition, if $r_i \in \{0,1\}^p$2 approximately maximizes $r_i \in \{0,1\}^p$3, then the induced completed-data law converges to the pseudo-true law, the conditional imputation law converges in total variation, and the posterior-mean imputation is risk-consistent. If the model is correctly specified and $r_i \in \{0,1\}^p$4, the pseudo-true target equals the true conditional distribution:

$r_i \in \{0,1\}^p$5

The assumptions listed for this result include compact sieve approximations for parameter spaces, uniform convergence of the profiled objective, vanishing optimization error, a unique law-level maximizer on an evaluation set, positivity of the observed-data marginal on that set, and bounded support for the missing coordinates when proving posterior-mean risk bounds. This suggests that the theorem is intended as a controlled asymptotic guarantee for the alternating Bayesian-generative procedure rather than as an unconditional guarantee under unrestricted model misspecification.

5. Empirical evaluation

The empirical study covers both synthetic and real-world tabular datasets, with emphasis on MNAR settings and on uncertainty calibration (Liu, 3 May 2026).

Dataset or benchmark	Size	Notes
Synthetic benchmark	$r_i \in \{0,1\}^p$6, $r_i \in \{0,1\}^p$7	Target block self-masked via MNAR mechanism
Wine	$r_i \in \{0,1\}^p$8, $r_i \in \{0,1\}^p$9	UCI dataset
Breast Cancer Wisconsin (Original)	$$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$0, $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$1	UCI dataset
Concrete	$$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$2, $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$3	UCI dataset
Gisette	$$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$4, $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$5	UCI dataset

For the synthetic benchmark, the sample sizes are $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$6 and the missing rates are $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$7. This setting has closed-form conditional truth, so both imputation error and uncertainty calibration can be evaluated exactly. On the real datasets, all features are standardized feature-wise before masking. The injected MNAR mask makes observation probability depend on the feature’s own value and on a nonlinear combination of the other features, so that larger or structurally extreme values are more likely to be missing.

MissBGM is compared with seven baselines: Mean, OT, ICE, miceforest, GAIN, ForestDiffusion, and TabCSDI. The evaluation metrics include RMSE on missing entries for point-imputation quality, and for uncertainty quality RMSE between estimated posterior standard deviation and oracle standard deviation, average width of predicted intervals, and Pearson and Spearman correlations between predicted interval widths and oracle widths. For uncertainty, the experiments use 1000 posterior imputations per missing entry.

The main reported results are as follows. On the synthetic benchmark, MissBGM achieves the lowest RMSE in 10 of 12 synthetic configurations. It generally outperforms mean imputation, OT, ICE, miceforest, GAIN, ForestDiffusion, and TabCSDI. A central pattern reported in the paper is that classical methods are competitive but lose because they do not model the missingness mechanism, while GAIN and TabCSDI often collapse to nearly degenerate uncertainty. MissBGM achieves the lowest RMSE on posterior standard deviation estimation; its intervals are somewhat conservative, but much more informative than the near-zero widths from some deep baselines.

On the real datasets, MissBGM obtains the best RMSE on all four real datasets. On Gisette, it reduces RMSE by about 10.2% relative to the best competing method. The paper also reports a scalability advantage in high-dimensional settings: some baselines fail to run on Gisette, whereas MissBGM scales to the high-dimensional case. Robustness is reported as dimensionality increases and as missingness rate increases, with especially strong behavior on the synthetic MNAR benchmark, where many methods degrade sharply. In running time, MissBGM is described as computationally practical: much faster than miceforest and diffusion-based methods, roughly comparable to GAIN, and slower than trivial methods like mean imputation.

An ablation study examines the mask weight $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$8. When $$p(X,R,Z,\theta,\phi) = \pi(\theta)\pi(\phi)\prod_{i=1}^n \pi(z_i)\,p_\theta(x_i\mid z_i)\,p_\phi(r_i\mid x_i),$$9, the method ignores the missingness mechanism; small $\pi(z_i)=\mathcal N(0,I_d).$0 is often best; and if $\pi(z_i)=\mathcal N(0,I_d).$1 is too large, performance can degrade because the mask starts to dominate the imputation. The default adopted in the paper is $\pi(z_i)=\mathcal N(0,I_d).$2.

6. Contributions, limitations, implementation, and nomenclature

The reported contributions of MissBGM are explicit joint modeling of data and missingness, Bayesian generative imputation with posterior imputations rather than only point estimates, alternating stochastic optimization plus HMC posterior sampling, a consistency result for convergence to a pseudo-true tempered target under regularity assumptions, and strong empirical performance across synthetic and real benchmarks (Liu, 3 May 2026).

The limitations noted in the paper are equally specific. Performance depends on correct or at least useful specification of the data and missingness models. Neural-network training and MCMC add computational cost relative to simple imputers. The consistency theorem is for the tempered pseudo-true target, not automatically the exact true posterior under misspecification. Future work suggested in the paper includes identifiability analysis via nonlinear ICA ideas and more advanced architectures for $\pi(z_i)=\mathcal N(0,I_d).$3 and $\pi(z_i)=\mathcal N(0,I_d).$4.

The implementation is open source at https://github.com/liuq-lab/MissBGM. It is implemented in Python and uses TensorFlow for model training and TensorFlow Probability for Bayesian neural networks and HMC-within-Gibbs. The software API example given is MissBGM(params, random_seed=42) with model.fit(...) and model.predict(...); the predict method returns posterior-mean imputations and posterior prediction intervals.

The name has a separate usage in the Fermi Gamma-ray Burst Monitor literature. In connection with "Background fitting of Fermi GBM observations" (Szécsi et al., 2013), “MissBGM” refers to the Direction Dependent Background Fitting background model for Fermi GBM observations rather than to missing-data imputation. That 2013 work is concerned with background fitting of GBM observations and models the background using direction-dependent variables derived from spacecraft motion. Accordingly, the term “MissBGM” is context-sensitive: in contemporary machine learning usage it denotes the missingness-aware Bayesian imputation framework, whereas in the Fermi GBM context it denotes a distinct background-fitting method.