Bayesian Federated Inference

CRAN_Status_Badge Project Status: Active - The project has reached a stable, usable state and is being actively developed. MIT license code size


Due to the limited size of the available data sets especially in rare diseases, it is sometimes challenging to identify the most relevant predictive features using multivariable statistical analysis. This issue may be resolved by combining data from multiple centers into one centralized location without sharing their data with each other, but doing so is difficult in reality because of privacy and security concerns.

To address these challenges, we developed and implemented a Bayesian Federated Inference (BFI) framework for multicenter data. It aims to leverage the statistical power of larger (combined) data sets without requiring all the data to be aggregated in one location. The BFI framework allows each center using their own local data to infer the optimal parameter values as well as additional features of the posterior parameter distribution to be able to gather more information which is not captured by alternative techniques. One of the benefit of BFI over alternative approaches is that, only one inference cycle across the centers is required in BFI.

An R package called BFI is created to perform Bayesian Federated Inference. The following instructions will install the development version of the BFI package to a computer. Python and SAS users can also apply the BFI methodology in their respective environments. For instructions, see here for Python and here for SAS.

Install R and RStudio

First, you need to install R and RStudio:

For more details about installing R and RStudio, see this page. If you need help learning R, see RStudio Education.

Install BFI package

In order to install the BFI package, invoke R or RStudio and then follow one of the following steps:

Installing from R

To install and load the BFI package directly from R, type the following (in Console)


Installing from Github

To install the BFI package directly from Github, you need to have the devtools package. So type the following:

if(!require(devtools)) {install.packages("devtools")}

and then load it by typing:


Next, install BFI as follows:

devtools::install_github("hassanpazira/BFI", dependencies = TRUE, build_vignettes = TRUE, force = TRUE)

The package can now be loaded into R and used by:



The latest version of the BFIpackage is 2.0.1. To check the current version of BFI installed in your R library, use:



The BFI package provides several functions, the most important of which are the following two main functions:

To access the R documentation for these functions, for example bfi(), enter the following command:

help(bfi, package = "BFI")  # without loading the BFI package
# or, equivalently, after loading the BFI package 


Let’s look at the following example to see how the BFI package can be used. For more examples and details look at the BFI vignette by typing

browseVignettes("BFI")  # to see all vignettes from the BFI package in an HTML browser.

or use vignette("BFI"), vignette("SAS") or vignette("Python") to see the BFI, SAS or Python vignettes separately in the Help tab of RStudio.

Now, we generate two independent (local) data sets from Gaussian distribution, and then apply the package to see how it works. First apply the function MAP.estimation() to each local data, and then apply the bfi() function to the aggregated results.

# y ~ Gaussian
# model assumptions:
p     <- 3                     # number of coefficients without intercept
theta <- c(1, rep(2, p), 1.5)  # regression coefficients (theta[1] is the intercept) and sigma2 = 1.5

# Data simulation for local center 1
n1   <- 30                                       # sample size of center 1
X1   <- data.frame(matrix(rnorm(n1 * p), n1, p)) # continuous variables
# linear predictor:
eta1 <- theta[1] + as.matrix(X1) %*% theta[2:4]
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1  <- gaussian()$linkinv(eta1)
y1   <- rnorm(n1, mu1, sd = sqrt(theta[5]))

# Data simulation for local center 2
n2   <- 50                                       # sample size of center 2
X2   <- data.frame(matrix(rnorm(n2 * p), n2, p)) # continuous variables
# linear predictor:
eta2 <- theta[1] + as.matrix(X2) %*% theta[2:4]
# inverse of the link function:
mu2  <- gaussian()$linkinv(eta2)
y2   <- rnorm(n2, mu2, sd = sqrt(theta[5]))

# Load the BFI package

# Inverse Covariance Matrix
# Creating the inverse covariance matrix for the Gaussian prior distribution:
Lambda <- inv.prior.cov(X1, lambda=0.05, family='gaussian') # the same for both centers

# MAP estimates at center 1
fit1       <- MAP.estimation(y1, X1, family='gaussian', Lambda)
theta_hat1 <- fit1$theta_hat # intercept and coefficient estimates
A_hat1     <- fit1$A_hat     # minus the curvature matrix

# MAP estimates at center 2
fit2       <- MAP.estimation(y2, X2, family='gaussian', Lambda)
theta_hat2 <- fit2$theta_hat
A_hat2     <- fit2$A_hat

# BFI at central center
A_hats     <- list(A_hat1, A_hat2)
theta_hats <- list(theta_hat1, theta_hat2)
bfi        <- bfi(theta_hats, A_hats, Lambda)
summary(bfi, cur_mat=TRUE)

# Stratified analysis
# Stratified analysis when 'intercept' varies across two centers:
newLambda1 <- inv.prior.cov(X1, lambda=c(0.1, 0.3), family='gaussian', stratified=TRUE, strat_par = 1)
# 'newLambda1' is used the prior for combined data and 'Lambda' is used the prior for locals
bfi1 <- bfi(theta_hats, A_hats, list(Lambda, newLambda1), stratified=TRUE, strat_par=1)
summary(bfi1, cur_mat=TRUE)

# Stratified analysis when 'sigma2' varies across two centers:
newLambda2 <- inv.prior.cov(X1, lambda=c(0.1, 0.3), family='gaussian', stratified=TRUE, strat_par = 2)
# 'newLambda2' is used the prior for combined data and 'Lambda' is used the prior for locals
bfi2 <- bfi(theta_hats, A_hats, list(Lambda, newLambda2), stratified=TRUE, strat_par=2)
summary(bfi2, cur_mat=TRUE)

# Stratified analysis when 'intercept' and 'sigma2' vary across 2 centers:
newLambda3 <- inv.prior.cov(X1, lambda=c(0.1, 0.2, 0.3), family='gaussian', stratified=TRUE, strat_par = c(1, 2))
# 'newLambda3' is used the prior for combined data and 'Lambda' is used the prior for locals
bfi3 <- bfi(theta_hats, A_hats, list(Lambda, newLambda3), stratified=TRUE, strat_par=1:2)
summary(bfi3, cur_mat=TRUE)


To cite BFI in publications, please use:



Here are some of technical papers of the package:


If you find any errors, have any suggestions, or would like to request that something be added, please file an issue at issue report or send an email to: