Post-Inference Model Chaining: A Problem Statement and a Model-based Bootstrap for Training End-to-End Digital Twins
Abstract
End-to-end digital twins can be constructed by fitting separate stage models and connecting them after inference. We investigate the Frequentist prediction properties of a commonly used Monte Carlo simulation scheme that forwards complete predictive draws through such a chain when measured stage outputs contain independent normal errors, but the latent stage means are deterministic. Analytical two-stage examples show persistent asymptotic interval and coverage inflation and, with quadratic propagation, mean bias dependent on the parameters to be estimated. Simulations of four- and eight-stage linear-quadratic univariate chains with 7, 15 or 30 training runs demonstrate pronounced input-dependent distortions. A diagnostic using error-free latent training predictors shows that the effect is mainly not attributable to errors in variables, but to the MCSim approach even showing underestimation of coverage. We propose a model-based bootstrap: an initial chain forwards fitted means, bootstrap chains are refitted on fixed-center bootstrap data while forwarding their mean prediction, and a target-stage future error enters a studentized prediction root. Across the investigated scenarios, relative mean bias is substantially reduced, and coverage is closer to the nominal 90%. At sample size 30, the eight-stage bootstrap has mean width ratios of 1.16-1.42 relative to the known-parameter normal reference, compared with 4.11-4.97 for observed-to-observed MCSim. These results support improved endpoint prediction in the tested setting. Further investigations of the bootstrap scheme in the supplement for a bivariate 6-stage process show similar satisfactory results.
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BibTeX
@misc{madenach2026postinference,
title = {{Post-Inference Model Chaining: A Problem Statement and a Model-based Bootstrap for Training End-to-End Digital Twins}},
author = {Lukas Madenach and Timo Bailer and Tobias Eilert},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05225},
eprint = {2611.05225},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05225},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Post-Inference Model Chaining: A Problem Statement and a Model-based Bootstrap for Training End-to-End Digital Twins
AU - Lukas Madenach
AU - Timo Bailer
AU - Tobias Eilert
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05225
UR - https://arxiv.org/abs/2611.05225
AB - End-to-end digital twins can be constructed by fitting separate stage models and connecting them after inference. We investigate the Frequentist prediction properties of a commonly used Monte Carlo simulation scheme that forwards complete predictive draws through such a chain when measured stage outputs contain independent normal errors, but the latent stage means are deterministic. Analytical two-stage examples show persistent asymptotic interval and coverage inflation and, with quadratic propagation, mean bias dependent on the parameters to be estimated. Simulations of four- and eight-stage linear-quadratic univariate chains with 7, 15 or 30 training runs demonstrate pronounced input-dependent distortions. A diagnostic using error-free latent training predictors shows that the effect is mainly not attributable to errors in variables, but to the MCSim approach even showing underestimation of coverage. We propose a model-based bootstrap: an initial chain forwards fitted means, bootstrap chains are refitted on fixed-center bootstrap data while forwarding their mean prediction, and a target-stage future error enters a studentized prediction root. Across the investigated scenarios, relative mean bias is substantially reduced, and coverage is closer to the nominal 90%. At sample size 30, the eight-stage bootstrap has mean width ratios of 1.16-1.42 relative to the known-parameter normal reference, compared with 4.11-4.97 for observed-to-observed MCSim. These results support improved endpoint prediction in the tested setting. Further investigations of the bootstrap scheme in the supplement for a bivariate 6-stage process show similar satisfactory results.
ER -