Bayesian conjugate analysis for federated statistical inference
Abstract
Abstract In many research settings, sufficiently large sample sizes can only be achieved by combining data from multiple collection sites. However, pooling individual participant data in a central server is often restricted due to regulatory constraints. Federated inference addresses this challenge by distributing the statistical analysis across local sites, allowing pooled inference in a central server using privacy-preserving summary statistics. Although federated inference methods exist in a frequentist framework, the full potential of Bayesian approaches has not yet been explored. Bayesian methods offer distinct advantages, including the ability to incorporate prior knowledge and perform predictive checks for model criticism. A recently published Bayesian method for federated inference relies on approximate solutions even in linear regression scenarios where exact solutions are available. We therefore propose a different approach to federated inference using Bayesian conjugate analysis (BCA), which is communication-efficient and mathematically convenient. For linear regression problems, BCA yields lossless parameter inference, that is, producing the same posterior distribution as if the pooled data had been analyzed. For problems where the parameters estimates are asymptotically normal (such as generalized linear models), BCA is equivalent to a multivariate fixed-effects meta-analysis, up to prior specification. We further show that BCA naturally lends itself to Reverse-Bayes analysis, which allows for computationally efficient predictive checks and identification of outlier sites. An implementation of BCA is available through the open-source confeR package (conjugate federated analysis in R). Our proposed framework thus facilitates privacy-preserving multi-center trials in a Bayesian setting.
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Bayesian conjugate analysis for federated statistical inference Supplementary Materials Peter Methys Degen1, Samuel Pawel1,2, and Leonhard Held1,2 1Center for Reproducible Science and Research Synthesis, University of Zurich, 8001, Zurich, Switzerland 2Epidemiology, Biostatistics and Prevention Institute, University of Zurich, 8001, Zurich, Switzerland December 12, 2025 Contents 1 Additional tables 1 2 Additional figures 5 1 Additional tables 1
Parameter Method Estimate Lower Upper Error (Intercept) Combined -1.627199 -2.129034 -1.182596 0.000000 (Intercept) BCA -1.520559 -1.998834 -1.042284 0.106640 (Intercept) ODAL -1.627273 -2.097037 -1.157509 0.000074 (Intercept) BFI -1.511768 -1.976339 -1.047196 0.115431 (Intercept) FE -1.520559 -1.998834 -1.042284 0.106640 (Intercept) REML -1.537118 -2.030847 -1.043390 0.090081 GCS Combined -2.000078 -2.504491 -1.564749 0.000000 GCS BCA -1.865543 -2.328175 -1.402911 0.134535 GCS ODAL -1.999469 -2.465103 -1.533835 0.000609 GCS BFI -1.854867 -2.302829 -1.406906 0.145210 GCS FE -1.865543 -2.328175 -1.402911 0.134535 GCS REML -1.890711 -2.370998 -1.410425 0.109366 ISS Combined 0.549925 0.189975 0.923115 0.000000 ISS BCA 0.538824 0.165286 0.912362 0.011101 ISS ODAL 0.550282 0.185730 0.914835 0.000358 ISS BFI 0.538832 0.170910 0.906754 0.011093 ISS FE 0.538824 0.165286 0.912362 0.011101 ISS REML 0.535889 0.143470 0.928307 0.014036 age Combined 1.371739 0.991038 1.801874 0.000000 age BCA 1.272259 0.872785 1.671732 0.099480 age ODAL 1.371581 0.969553 1.773609 0.000158 age BFI 1.263880 0.874844 1.652917 0.107859 age FE 1.272259 0.872785 1.671732 0.099480 age REML 1.281000 0.862652 1.699348 0.090739 sex Combined -0.335881 -1.133824 0.439742 0.000000 sex BCA -0.362788 -1.160957 0.435381 0.026907 sex ODAL -0.335728 -1.118094 0.446638 0.000153 sex BFI -0.360434 -1.141489 0.420622 0.024553 sex FE -0.362788 -1.160957 0.435381 0.026907 sex REML -0.366746 -1.175479 0.441987 0.030865 Table 1: Parameter estimates for the homogeneous trauma data set with a global intercept. The “Error” column shows the difference of the estimate to the ground truth from the combined data 2
Parameter Method Estimate Lower Upper Error σ2Combined 0.911882 0.836773 0.997632 0.000000 σ2BCA 0.911882 0.836773 0.997632 0.000000 σ2DLMM 0.911882 NaN NaN 0.000000 σ2BFI 0.537118 0.493324 0.580913 0.374764 σ2FE 0.615467 0.561634 0.677470 0.296415 σ2REML 0.615467 0.561634 0.677470 0.296415 (Intercept) Combined 0.332377 0.203233 0.461521 0.000000 (Intercept) BCA 0.332377 0.203233 0.461521 0.000000 (Intercept) DLMM 0.332377 0.203233 0.461521 0.000000 (Intercept) BFI 0.522507 0.438681 0.606333 0.190130 (Intercept) FE 0.517511 0.427672 0.607350 0.185134 (Intercept) REML 0.482095 0.378248 0.585942 0.149718 age Combined 0.232906 0.130581 0.335230 0.000000 age BCA 0.232906 0.130581 0.335230 0.000000 age DLMM 0.232906 0.130581 0.335230 0.000000 age BFI 0.263603 0.197426 0.329780 0.030697 age FE 0.262817 0.191914 0.333720 0.029911 age REML 0.227934 0.140189 0.315678 0.004972 experience Combined -0.352280 -0.454675 -0.249886 0.000000 experience BCA -0.352280 -0.454675 -0.249886 0.000000 experience DLMM -0.352280 -0.454675 -0.249886 0.000000 experience BFI -0.385664 -0.452027 -0.319302 0.033384 experience FE -0.384810 -0.455878 -0.313741 0.032530 experience REML -0.393977 -0.482411 -0.305543 0.041697 gender Combined -0.503225 -0.637469 -0.368980 0.000000 gender BCA -0.503225 -0.637469 -0.368980 0.000000 gender DLMM -0.503225 -0.637469 -0.368980 0.000000 gender BFI -0.502971 -0.589991 -0.415951 0.000254 gender FE -0.501942 -0.595190 -0.408693 0.001283 gender REML -0.513292 -0.620148 -0.406435 0.010067 wardtype Combined 0.074836 -0.043595 0.193268 0.000000 wardtype BCA 0.074836 -0.043595 0.193268 0.000000 wardtype DLMM 0.074836 -0.043595 0.193268 0.000000 wardtype BFI -0.011112 -0.087870 0.065647 0.085948 wardtype FE -0.007565 -0.089796 0.074666 0.082402 wardtype REML -0.025546 -0.122151 0.071058 0.100383 Table 2: Parameter estimates for the nurses data set with a global intercept. The “Error” column shows the difference of the estimate to the ground truth from the combined data 3
Parameter Method Estimate Lower Upper Error σ2Combined 0.631774 0.579163 0.691931 0.000000 σ2BCA 0.631774 0.579163 0.691931 0.000000 σ2DLMM 0.631774 NaN NaN 0.000000 σ2BFI 0.580202 0.536411 0.623992 0.051572 σ2FE 0.645030 0.589348 0.709036 0.013257 σ2REML 0.645030 0.589348 0.709036 0.013257 age Combined 0.247045 0.160765 0.333326 0.000000 age BCA 0.247045 0.160765 0.333326 0.000000 age DLMM 0.247045 0.160765 0.333326 0.000000 age BFI 0.267991 0.198146 0.337836 0.020946 age FE 0.267580 0.193812 0.341348 0.020535 age REML 0.267580 0.193812 0.341348 0.020535 experience Combined -0.357006 -0.443448 -0.270565 0.000000 experience BCA -0.357006 -0.443448 -0.270565 0.000000 experience DLMM -0.357006 -0.443448 -0.270565 0.000000 experience BFI -0.363841 -0.434414 -0.293268 0.006835 experience FE -0.363390 -0.437901 -0.288879 0.006383 experience REML -0.363390 -0.437901 -0.288879 0.006383 gender Combined -0.473714 -0.586745 -0.360683 0.000000 gender BCA -0.473714 -0.586745 -0.360683 0.000000 gender DLMM -0.473714 -0.586745 -0.360683 0.000000 gender BFI -0.451465 -0.542849 -0.360081 0.022249 gender FE -0.452495 -0.549160 -0.355830 0.021219 gender REML -0.452495 -0.549160 -0.355830 0.021219 Table 3: Parameter estimates for the nurses data set with 25 local intercepts (not shown). The “Error” column shows the difference of the estimate to the ground truth from the combined data 4
2 Additional figures 1.0 0.5 0.0 0.5 1.0 Intercept (Combined) 1.0 0.5 0.0 0.5 1.0 1.5 Intercept (Federated) A BCA BFI DLMM Local BCA BFI DLMM Local 10 12 10 10 10 8 10 6 10 4 10 2 100 Absolute Error B Figure 1: A: Site-specific estimates for the 25 intercepts from the nurses data set. The x-axis shows the fixed-effect intercepts estimated with the combined data. The y-axis shows fixed-effect estimates obtained from federated data with different methods: BCA, BFI, DLMM, as well as the local estimates obtained from each site independently. The dashed shows the identity function y=x.B: Absolute error of intercept estimates, relative to the combined data. Each box plot contains the estimates from 25 intercepts 5
0.02 0.16 0.30 0.72 0.77 0.78 0.98 1.07 1.87 2.07 2.33 3.09 3.26 3.42 4.00 6.67 7.36 7.85 8.78 10.29 12.56 14.70 19.04 45.34 63.60 sBox (Intercept) age experience gender wardtype −2 −1 0 1 2 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 −1.5 −1.0 −0.5 0.0 0.5 −1 0 1 9 21 23 19 15 14 13 25 10 5 6 22 1 4 11 24 8 16 20 7 18 12 2 3 17 Effect estimate (95 % CI) Site Figure 2: Forest plot of local site estimates for all covariates (plus a global intercept) for the nurses data set. The red diamonds show the federated BCA estimates, with the CI given by the width of the diamonds. Values of sBox =−log2pBox are printed on the right. Values that pass a Bonferroni-corrected significance threshold of pBox <0.05/25 are shown in bold 6
0.15 0.23 0.31 0.39 0.48 0.51 0.81 0.82 1.07 1.12 1.17 1.31 1.33 2.23 2.61 2.64 2.66 3.00 3.03 3.17 3.66 4.39 4.67 6.26 6.67 sBox age experience gender −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 −1.5 −1.0 −0.5 0.0 0.5 5 12 11 6 23 1 13 4 14 8 16 18 21 15 7 10 24 9 3 17 2 22 19 20 25 Effect estimate (95 % CI) Site Figure 3: Forest plot of local site estimates for all covariates for the nurses data set using site-specific intercepts (not shown). The red diamonds show the federated BCA estimates, with the CI given by the width of the diamonds. Values of sBox =−log2pBox are printed on the right. Values that pass a Bonferronicorrected significance threshold of pBox <0.05/25 are shown in bold 7