# mpost.euc: Median Posterior for Subset Posterior Samples in Euclidean... In SBmedian: Scalable Bayes with Median of Subset Posteriors

## Description

mpost.euc is a general framework to merge multiple empirical measures Q_1,Q_2,…,Q_M \subset R^p from independent subset of data by finding a median

\hat{Q} = \textrm{argmin}_Q ∑_{m=1}^M d(Q,Q_m)

where Q is a weighted combination and d(P_1,P_2) is distance in RKHS between two empirical measures P_1 and P_2. As in the references, we use RBF kernel with bandwidth parameter σ.

## Usage

 1 2 3 4 5 6 7 mpost.euc( splist, sigma = 0.1, maxiter = 121, abstol = 1e-06, show.progress = FALSE ) 

## Arguments

 splist a list of length M containing vectors or matrices of univariate or multivariate subset posterior samples respectively. sigma bandwidth parameter for RBF kernel. maxiter maximum number of iterations for Weiszfeld algorithm. abstol stopping criterion for Weiszfeld algorithm. show.progress a logical; TRUE to show iteration mark, FALSE otherwise.

## Value

a named list containing:

med.atoms

a vector or matrix of all atoms aggregated.

med.weights

a weight vector that sums to 1 corresponding to med.atoms.

weiszfeld.weights

a weight for M subset posteriors.

weiszfeld.history

updated parameter values. Each row is for iteration, while columns are weights corresponding to weiszfeld.weights.

## References

\insertRef

minsker_scalable_2014SBmedian

\insertRef

minsker_robust_2017SBmedian

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ## Median Posteior from 2-D Gaussian Samples # Step 1. let's build a list of atoms whose numbers differ set.seed(8128) # for reproducible results mydata = list() mydata[[1]] = cbind(rnorm(96, mean= 1), rnorm(96, mean= 1)) mydata[[2]] = cbind(rnorm(78, mean=-1), rnorm(78, mean= 0)) mydata[[3]] = cbind(rnorm(65, mean=-1), rnorm(65, mean= 1)) mydata[[4]] = cbind(rnorm(77, mean= 2), rnorm(77, mean=-1)) # Step 2. Let's run the algorithm myrun = mpost.euc(mydata, show.progress=TRUE) # Step 3. Visualize # 3-1. show subset posterior samples opar <- par(no.readonly=TRUE) par(mfrow=c(2,3), no.readonly=TRUE) for (i in 1:4){ plot(mydata[[i]], cex=0.5, col=(i+1), pch=19, xlab="", ylab="", main=paste("subset",i), xlim=c(-4,4), ylim=c(-3,3)) } # 3-2. 250 median posterior samples via importance sampling id250 = base::sample(1:nrow(myrun$med.atoms), 250, prob=myrun$med.weights, replace=TRUE) sp250 = myrun$med.atoms[id250,] plot(sp250, cex=0.5, pch=19, xlab="", ylab="", xlim=c(-4,4), ylim=c(-3,3), main="median samples") # 3-3. convergence over iterations matplot(myrun$weiszfeld.history, xlab="iteration", ylab="value", type="b", main="convergence of weights") par(opar) 

SBmedian documentation built on Aug. 16, 2021, 9:07 a.m.