An Implementation of Rubin's (1981) Bayesian Bootstrap.

bayesboot: Easy Bayesian Bootstrap in R

The bayesboot package implements a function bayesboot that performs the Bayesian bootstrap introduced by Rubin (1981). The implementation can both handle summary statistics that works on a weighted version of the data or that works on a resampled data set.

bayesboot is available on CRAN and can be installed in the usual way:

install.packages("bayesboot")

A simple example

Here is a Bayesian bootstrap analysis of the mean height of the last ten American presidents:

# Heights of the last ten American presidents in cm (Kennedy to Obama).
heights <- c(183, 192, 182, 183, 177, 185, 188, 188, 182, 185)

library(bayesboot)
b1 <- bayesboot(heights, mean)

The resulting posterior distribution in b1 can now be plotted and summaryized:

summary(b1)
## Bayesian bootstrap
## 
## Number of posterior draws: 4000 
## 
## Summary of the posterior (with 95% Highest Density Intervals):
##  statistic     mean       sd  hdi.low hdi.high
##         V1 184.4979 1.154456 182.3328  186.909
## 
## Quantiles:
##  statistic    q2.5%    q25%   median     q75%   q97.5%
##         V1 182.2621 183.742 184.4664 185.2378 186.8737
## 
## Call:
##  bayesboot(data = heights, statistic = mean)
plot(b1)

While it is possible to use a summary statistic that works on a resample of the original data, it is more efficient if it's possible to use a summary statistic that works on a reweighting of the original dataset. Instead of using mean above it would be better to use weighted.mean like this:

b2 <- bayesboot(heights, weighted.mean, use.weights = TRUE)

The result of a call to bayesboot will always result in a data.frame with one column per dimension of the summary statistic. If the summary statistic does not return a named vector the columns will be called V1, V2, etc. The result of a bayesboot call can be further inspected and post processed. For example:

# Given the model and the data, this is the probability that the mean
# heights of American presidents is above the mean heights of
# American males as given by www.cdc.gov/nchs/data/series/sr_11/sr11_252.pdf
mean(c(b2[,1] > 175.9, TRUE, FALSE))
## [1] 0.9997501

Comparing two groups

If we want to compare the means of two groups, we will have to call bayesboot twice with each dataset and then use the resulting samples to calculate the posterior difference. For example, let's say we have the heights of the opponents that lost to the presidents in height the first time those presidents were elected. Now we are interested in comparing the mean height of American presidents with the mean height of presidential candidates that lost.

# The heights of oponents of American presidents (first time they were elected).
# From Richard Nixon to John McCain
heights_opponents <- c(182, 180, 180, 183, 177, 173, 188, 185, 175)

# Running the Bayesian bootstrap for both datasets
b_presidents <- bayesboot(heights, weighted.mean, use.weights = TRUE)
b_opponents  <- bayesboot(heights_opponents, weighted.mean, use.weights = TRUE)

# Calculating the posterior difference
# (here converting back to a bayesboot object for pretty plotting)
b_diff <- as.bayesboot(b_presidents - b_opponents)
plot(b_diff)

So there is some evidence that loosing opponents could be shorter. (Though, I must add that it is quite unclear what the purpose really is with analyzing the heights of presidents and opponents...)

A more advanced example

A slightly more complicated example, where we do Bayesian bootstrap analysis of LOESS regression applied to the cars dataset on the speed of cars and the resulting distance it takes to stop. The loess function returns, among other things, a vector of fittedy values, one value for each x value in the data. These y values define the smoothed LOESS line and is what you would usually plot after having fitted a LOESS. Now we want to use the Bayesian bootstrap to gauge the uncertainty in the LOESS line. As the loess function accepts weighted data, we'll simply create a function that takes the data with weights and returns the fittedy values. We'll then plug that function into bayesboot:

boot_fn <- function(cars, weights) {
  loess(dist ~ speed, cars, weights = weights)$fitted
}

bb_loess <- bayesboot(cars, boot_fn, use.weights = TRUE)

To plot this takes a couple of lines more:

# Plotting the data
plot(cars$speed, cars$dist, pch = 20, col = "tomato4", xlab = "Car speed in mph", 
     ylab = "Stopping distance in ft", main = "Speed and Stopping distances of Cars")

# Plotting a scatter of Bootstrapped LOESS lines to represent the uncertainty.
for(i in sample(nrow(bb_loess), 20)) {
  lines(cars$speed, bb_loess[i,], col = "gray")
}
# Finally plotting the posterior mean LOESS line
lines(cars$speed, colMeans(bb_loess, na.rm = TRUE), type ="l",
      col = "tomato", lwd = 4)

More information

For more information on the Bayesian bootstrap see Rubin's (1981) original paper and my blog post The Non-parametric Bootstrap as a Bayesian Model. The implementation of bayesboot is similar to the function outlined in the blog post Easy Bayesian Bootstrap in R, but the interface is slightly different.

References

Rubin, D. B. (1981). The Bayesian bootstrap. The annals of statistics, 9(1), 130--134. link to paper.