LOCO-FL: Scaling and Reference Choice for Bayesian Structural Analysis of Between-Center Heterogeneity in Federated Learning with Markov Boundaries

András Millinghoffer, Júlia Kisida, Andrea Valek, Miklós Szabó, Péter Antal
Proceedings of the 13th International Conference on Probabilistic Graphical Models, PMLR 343:646-662, 2026.

Abstract

Multicenter quality-improvement programs still struggle to explain between-center heterogeneity using outcome measures alone. We investigate a Bayesian network method that focuses on Markov boundaries to characterize this heterogeneity at the structural level of risk models with multiple outcomes. Working with the full structure posterior rather than a point estimate allows the study of a phenomenon that point-identification learners overlook: under proper regularization, the resolvable Markov-boundary complexity grows with sample size. A naive comparison across centers, therefore, mixes sample-size induced model properties with genuine structural differences. We therefore study a global reference versus leave-one-center-out (LOCO) references together with sample-size correction, and show that both can be analyzed within a single reweighting of the per-family BD sufficient statistics, achieved exactly by adding the $\beta$-scaled LOCO counts as pseudocounts to the score. We investigate augmenting small centers to a larger, common effective size using the LOCO sample. We present two exact, privacy-preserving (PP) Bayesian federated learning schemes: a single global-reference scheme and a LOCO-reference scheme (LOCO-FL). We show that the LOCO-FL scheme supports per-center reference comparison and per-center scaling under the same institutional-privacy guarantees. Because the LOCO aggregate excludes a center’s own data, the same LOCO counts can also serve as an informative prior for that center’s local inference, without self-inclusion. We choose the scale of this prior empirically, as the value that most reduces the posterior spread, which reveals a trade-off between variance reduction and loss of the between-center signal. On synthetic data with known between-center differences, the method recovers these differences above the within-posterior spread. We report results for an eight-center Hungarian very preterm cohort (2{,}939 patients), measuring heterogeneity by posterior Jaccard distances between LOCO and global references. Results confirm the need to scale to a common effective sample size, the utility of a single global reference for coarse comparison, and that an empirically scaled LOCO prior reduces posterior spread.

Cite this Paper

BibTeX
@InProceedings{pmlr-v343-millinghoffer26a, title = {{LOCO-FL}: Scaling and Reference Choice for {Bayesian} Structural Analysis of Between-Center Heterogeneity in Federated Learning with {Markov} Boundaries}, author = {Millinghoffer, Andr\'as and Kisida, J\'ulia and Valek, Andrea and Szab\'o, Mikl\'os and Antal, P\'eter}, booktitle = {Proceedings of the 13th International Conference on Probabilistic Graphical Models}, pages = {646--662}, year = {2026}, editor = {Camps-Valls, Gustau and Leonelli, Manuele and Varando, Gherardo}, volume = {343}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v343/main/assets/millinghoffer26a/millinghoffer26a.pdf}, url = {https://proceedings.mlr.press/v343/millinghoffer26a.html}, abstract = {Multicenter quality-improvement programs still struggle to explain between-center heterogeneity using outcome measures alone. We investigate a Bayesian network method that focuses on Markov boundaries to characterize this heterogeneity at the structural level of risk models with multiple outcomes. Working with the full structure posterior rather than a point estimate allows the study of a phenomenon that point-identification learners overlook: under proper regularization, the resolvable Markov-boundary complexity grows with sample size. A naive comparison across centers, therefore, mixes sample-size induced model properties with genuine structural differences. We therefore study a global reference versus leave-one-center-out (LOCO) references together with sample-size correction, and show that both can be analyzed within a single reweighting of the per-family BD sufficient statistics, achieved exactly by adding the $\beta$-scaled LOCO counts as pseudocounts to the score. We investigate augmenting small centers to a larger, common effective size using the LOCO sample. We present two exact, privacy-preserving (PP) Bayesian federated learning schemes: a single global-reference scheme and a LOCO-reference scheme (LOCO-FL). We show that the LOCO-FL scheme supports per-center reference comparison and per-center scaling under the same institutional-privacy guarantees. Because the LOCO aggregate excludes a center’s own data, the same LOCO counts can also serve as an informative prior for that center’s local inference, without self-inclusion. We choose the scale of this prior empirically, as the value that most reduces the posterior spread, which reveals a trade-off between variance reduction and loss of the between-center signal. On synthetic data with known between-center differences, the method recovers these differences above the within-posterior spread. We report results for an eight-center Hungarian very preterm cohort (2{,}939 patients), measuring heterogeneity by posterior Jaccard distances between LOCO and global references. Results confirm the need to scale to a common effective sample size, the utility of a single global reference for coarse comparison, and that an empirically scaled LOCO prior reduces posterior spread.} }
Endnote
%0 Conference Paper %T LOCO-FL: Scaling and Reference Choice for Bayesian Structural Analysis of Between-Center Heterogeneity in Federated Learning with Markov Boundaries %A András Millinghoffer %A Júlia Kisida %A Andrea Valek %A Miklós Szabó %A Péter Antal %B Proceedings of the 13th International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2026 %E Gustau Camps-Valls %E Manuele Leonelli %E Gherardo Varando %F pmlr-v343-millinghoffer26a %I PMLR %P 646--662 %U https://proceedings.mlr.press/v343/millinghoffer26a.html %V 343 %X Multicenter quality-improvement programs still struggle to explain between-center heterogeneity using outcome measures alone. We investigate a Bayesian network method that focuses on Markov boundaries to characterize this heterogeneity at the structural level of risk models with multiple outcomes. Working with the full structure posterior rather than a point estimate allows the study of a phenomenon that point-identification learners overlook: under proper regularization, the resolvable Markov-boundary complexity grows with sample size. A naive comparison across centers, therefore, mixes sample-size induced model properties with genuine structural differences. We therefore study a global reference versus leave-one-center-out (LOCO) references together with sample-size correction, and show that both can be analyzed within a single reweighting of the per-family BD sufficient statistics, achieved exactly by adding the $\beta$-scaled LOCO counts as pseudocounts to the score. We investigate augmenting small centers to a larger, common effective size using the LOCO sample. We present two exact, privacy-preserving (PP) Bayesian federated learning schemes: a single global-reference scheme and a LOCO-reference scheme (LOCO-FL). We show that the LOCO-FL scheme supports per-center reference comparison and per-center scaling under the same institutional-privacy guarantees. Because the LOCO aggregate excludes a center’s own data, the same LOCO counts can also serve as an informative prior for that center’s local inference, without self-inclusion. We choose the scale of this prior empirically, as the value that most reduces the posterior spread, which reveals a trade-off between variance reduction and loss of the between-center signal. On synthetic data with known between-center differences, the method recovers these differences above the within-posterior spread. We report results for an eight-center Hungarian very preterm cohort (2{,}939 patients), measuring heterogeneity by posterior Jaccard distances between LOCO and global references. Results confirm the need to scale to a common effective sample size, the utility of a single global reference for coarse comparison, and that an empirically scaled LOCO prior reduces posterior spread.
APA
Millinghoffer, A., Kisida, J., Valek, A., Szabó, M. & Antal, P.. (2026). LOCO-FL: Scaling and Reference Choice for Bayesian Structural Analysis of Between-Center Heterogeneity in Federated Learning with Markov Boundaries. Proceedings of the 13th International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 343:646-662 Available from https://proceedings.mlr.press/v343/millinghoffer26a.html.

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