tBayes-MICE: A Bayesian Approach to Multiple Imputation for Time Series Data

Amuche Ibenegbu; Pierre Lafaye de Micheaux; Rohitash Chandra

stat.MLMar 28, 2026

tBayes-MICE: A Bayesian Approach to Multiple Imputation for Time Series Data

Authors: Amuche Ibenegbu, Pierre Lafaye de Micheaux, Rohitash Chandra

Organizations: Transitional Artificial Intelligence Research Group School of Mathematics and Statistics UNSW Sydney Australia · School of Mathematics and Statistics UNSW Sydney Australia

Abstract

Time-series analysis is often affected by missing data, a common problem across several fields, including healthcare and environmental monitoring. Multiple Imputation by Chained Equations (MICE) has been prominent for imputing missing values through "fully conditional specification". We extend MICE using the Bayesian framework (tBayes-MICE), utilising Bayesian inference to impute missing values via Markov Chain Monte Carlo (MCMC) sampling to account for uncertainty in MICE model parameters and imputed values. We also include temporally informed initialisation and time-lagged features in the model to respect the sequential nature of time-series data. We evaluate the tBayes-MICE method using two real-world datasets (AirQuality and PhysioNet), and using both the Random Walk Metropolis (RWM) and the Metropolis-Adjusted Langevin Algorithm (MALA) samplers. Our results demonstrate that tBayes-MICE reduces imputation errors relative to the baseline methods over all variables and accounts for uncertainty in the imputation process, thereby providing a more accurate measure of imputation error. We also found that MALA mixed better than RWM across most variables, achieving comparable accuracy while providing more consistent posterior exploration. Overall, these findings suggest that the tBayes-MICE framework represents a practical and efficient approach to time-series imputation, balancing increased accuracy with meaningful quantification of uncertainty in various environmental and clinical settings.

Figures & tables

Figure 1: Time-lagged imputation mechanism in univariate MICE, where a missing value at time index $t$ is imputed by conditioning on observed neighbouring values from past and future lags.

X_{t,j}=f_{j}(\bm{Z}_{t,-j}^{\top}\theta_{j})+\varepsilon_{t,j},\quad\varepsilon_{t,j}\sim\mathcal{N}(0,\tau_{j}^{2})

Algorithm 2 tBayes-MICE for Multivariate Time Series with MCMC sampling.

Figure 2: The proposed tBayes-MICE imputation framework. The framework consists of temporal pattern detection, placeholder initialisation, MICE loop with lagged predictors and Bayesian modelling, while MCMC (RWM or MALA) provides posterior sampling and parameter updates. The final imputation is generated via posterior predictive draws.

Method	RMSE	MAE	NMAE	NRMSE
MICE (Time-Lagged)	7.8500	5.0708	0.6005	0.9296
Linear Interpolation	8.0579	5.2240	0.6186	0.9542
KNN ( $k{=}5$ )	8.4580	5.3885	0.6381	1.0016
Mean Imputation	8.4580	5.3885	0.6381	1.0016
Median Imputation	8.5415	5.1430	0.6090	1.0115
Seasonal Decomposition	8.7597	5.2980	0.6274	1.0373

Table 1: Imputation performance comparison of univariate time-lagged MICE (lag = 1) with established time series baseline methods on a representative variable (WBC) from the PhysioNet dataset . We report RMSE, MAE, NMAE, and NRMSE, Boldface highlights the best value among the methods.

Panel of Figure 3 — Figure 3: Convergence diagnostics for $\tau^{2}$ (trace, marginal density, and ACF) across two chains and two samplers for one of the variables ( HC0 3 ) from the physionet dataset.

Table 2: Convergence summaries for the bicarbonate ( HCO 3 ) variable from the PhysioNet dataset using tBayes-MICE. Results are shown for two MCMC samplers: MALA (left) and RWM (right). The table reports posterior means, standard deviations (SD), 94% highest density intervals (HDIs), effective sample sizes (ESS), and

\hat{R}

statistics.

		RWM		MALA
Variable	Method	NMAE	NRMSE	NMAE	NRMSE
CO(GT)	MICE	0.2622 (0.0071)	0.3376 (0.0083)	0.2622 (0.0071)	0.3376 (0.0083)
	BRITS	0.2646 (0.0473)	0.3870 (0.0653)	0.2646 (0.0473)	0.3870 (0.0653)
	tBayes-MICE	0.1349 (0.0013)	0.1788 ( 0.0017)	0.1346 (0.0011)	0.1788 (0.0015)
PT08.S1(CO)	MICE	0.3811 (0.0097)	0.4881 (0.0125)	0.3811 (0.0097)	0.4881 (0.0125)
	BRITS	0.3418 (0.0240)	0.6153 (0.0252)	0.3418 (0.0240)	0.6153 (0.0252)

Table 3: Imputation performance of MICE, BRITS and tBayes-MICE under two MCMC samplers (RWM/MALA). Metrics are NMAE and NRMSE (mean, standard deviation) across 30 experimental runs for six AirQuality variables. Boldface highlights the best value per row.

		RWM		MALA
Variable	Method	NMAE	NRMSE	NMAE	NRMSE
HCO3	MICE	1.1904 (0.0110)	1.4997 (0.0128)	1.1904 (0.0110)	1.4997 (0.0128)
	BRITS	0.7535 (0.0039)	0.9941 (0.0049)	0.7535 (0.0039)	0.9941 (0.0049)
	tBayes-MICE	0.7189 (0.0018)	0.9375 (0.0021)	0.7198 (0.0019)	0.9380 (0.0022)
Na	MICE	1.1467 (0.0098)	1.4549 (0.0122)	1.1467 (0.0098)	1.4549 (0.0122)
	BRITS	0.7732 (0.0023)	1.5002 (0.0018)	0.7732 (0.0023)	1.5002 (0.0018)

Table 4: Imputation performance of MICE, BRITS and tBayes-MICE under two MCMC samplers (RWM/MALA). Metrics are NMAE and NRMSE (mean, standard deviation) across 30 experimental runs for four PhysioNet variables. Boldface highlights the best value per row.

Panel of Figure 4 — Figure 4: Prediction accuracy comparison for the CO(GT) variable from the AirQuality dataset (top) and the HCO 3 variable from the physioNet dataset (bottom). Each subplot panel shows:(top-left) predicted versus true values, (top-right) residuals against the true values, (bottom-left) sorted true values with corresponding model predictions, and (bottom-right) box plots of absolute errors.

Panel of Figure 5 — Figure 5: Imputation error patterns over time index of each method across different datasets, with the AirQuality CO(GT) variable shown in the top panel and the PhysioNet HCO 3 variable shown in the bottom panel

Panel of Figure 6 — Figure 6: Posterior predictive performance of the proposed tBayes-MICE model over time for the environmental CO(GT) variable (top row) and the clinical Na variable (bottom row), shown across training(left) and test(right) sets. The blue lines represent observed values, the red lines indicate posterior predictive means, and the shaded regions denote the 95% credible intervals.

Figure 7: Comparative imputation accuracy and distribution across runs (box plot) for deterministic and Bayesian methods on the PhsyioNet dataset using NRMSE

Table 5: Convergence summaries for the CO(GT) variable from the AirQuality dataset using tBayes-MICE. Results are shown for two MCMC samplers: MALA (left) and RWM (right). The table reports posterior means, standard deviations (SD), 94% highest density intervals (HDIs), effective sample sizes (ESS), and

\hat{R}

statistics.

	AirQuality			PhysioNet
Method	Variable	NMAE	NRMSE	Variable	NMAE	NRMSE
tBayes-MICE-V1	CO(GT)	0.1346 (0.0011 )	0.1783 (0.0016)	HCO 3	0.7196 (0.0020 )	0.9381 (0.0024)
	NMHC(GT)	0.2209 (0.0025)	0.3464 (0.0029)	Na	0.7734 (0.0019)	1.0263 (0.0028)
	C 6 H 6 (GT)	0.1058 (0.0005)	0.1556 (0.0015)	Platelet	0.6585 (0.0021)	0.9443 (0.0021)
	T	0.1345 (0.0016)	0.1859 (0.0018)	WBC	0.6200 (0.0024)	0.9014 (0.0022)
tBayes-MICE-V2	CO(GT)	0.1349 (0.0013)	0.1788 (0.0017)	HCO 3	0.7189 0.0018)	0.9375 (0.0021)

Table 6: Ablation study comparing tBayes-MICE-V1 and tBayes-MICE-V2 on the AirQuality and PhysioNet datasets under the RWM sampler. Metrics are NMAE and NRMSE (mean, standard deviation) across 30 runs. Boldface indicates the better-performing variant for each variable within each dataset.

Time Series Imputation

Time series imputation reconstructs missing observations in sequential data using temporal, cross-variable, and domain-specific patterns.

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