# BayesianBootstrap: The Bayesian Bootstrap In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

This function performs the Bayesian bootstrap of Rubin (1981), returning either bootstrapped weights or statistics.

## Usage

 `1` ```BayesianBootstrap(X, n=1000, Method="weights", Status=NULL) ```

## Arguments

 `X` This is a vector or matrix of data. When a matrix is supplied, sampling is based on the first column. `n` This is the number of bootstrapped replications. `Method` When `Method="weights"` (which is the default), a matrix of row weights is returned. Otherwise, a function is accepted. The function specifies the statistic to be bootstrapped. The first argument of the function should be a matrix of data, and the second argument should be a vector of weights. `Status` This determines the periodicity of status messages. When `Status=100`, for example, a status message is displayed every 100 replications. Otherwise, `Status` defaults to `NULL`, and status messages are not displayed.

## Details

The term, ‘bootstrap’, comes from the German novel Adventures of Baron Munchausen by Rudolph Raspe, in which the hero saves himself from drowning by pulling on his own bootstraps. The idea of the statistical bootstrap is to evaluate properties of an estimator through the empirical, rather than theoretical, CDF.

Rubin (1981) introduced the Bayesian bootstrap. In contrast to the frequentist bootstrap which simulates the sampling distribution of a statistic estimating a parameter, the Bayesian bootstrap simulates the posterior distribution.

The data, X, are assumed to be independent and identically distributed (IID), and to be a representative sample of the larger (bootstrapped) population. Given that the data has N rows in one bootstrap replication, the row weights are sampled from a Dirichlet distribution with all N concentration parameters equal to 1 (a uniform distribution over an open standard N-1 simplex). The distributions of a parameter inferred from considering many samples of weights are interpretable as posterior distributions on that parameter.

The Bayesian bootstrap is useful for estimating marginal posterior covariance and standard deviations for the posterior modes of `LaplaceApproximation`, especially when the model dimension (the number of parameters) is large enough that estimating the `Hessian` matrix of second partial derivatives is too computationally demanding.

Just as with the frequentist bootstrap, inappropriate use of the Bayesian bootstrap can lead to inappropriate inferences. The Bayesian bootstrap violates the likelihood principle, because the evaluation of a statistic of interest depends on data sets other than the observed data set. For more information on the likelihood principle, see https://web.archive.org/web/20150213002158/http://www.bayesian-inference.com/likelihood#likelihoodprinciple.

The `BayesianBootstrap` function has many uses, including creating test statistics on the population data given the observed data (supported here), imputation (with this variation: `ABB`), validation, and more.

## Value

When `Method="weights"`, this function returns a S x J matrix of weights, where the number of rows N is equal to the number of rows in `X`.

For statistics, a matrix or array is returned, depending on the number of dimensions. The replicates are indexed by row in a matrix or in the first dimension of the array.

## Author(s)

Bogumil Kaminski, bkamins@sgh.waw.pl and Statisticat, LLC.

## References

Rubin, D.B. (1981). "The Bayesian Bootstrap". The Annals of Statistics, 9(1), p. 130–134.

`ABB`, `Hessian`, `LaplaceApproximation`, and `LaplacesDemon`.
 ``` 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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53``` ```library(LaplacesDemon) #Example 1: Samples x <- 1:2 BB <- BayesianBootstrap(X=x, n=100, Method="weights"); BB #Example 2: Mean, Univariate x <- 1:2 BB <- BayesianBootstrap(X=x, n=100, Method=weighted.mean); BB #Example 3: Mean, Multivariate data(demonsnacks) BB <- BayesianBootstrap(X=demonsnacks, n=100, Method=function(x,w) apply(x, 2, weighted.mean, w=w)); BB #Example 4: Correlation dye <- c(1.15, 1.70, 1.42, 1.38, 2.80, 4.70, 4.80, 1.41, 3.90) efp <- c(1.38, 1.72, 1.59, 1.47, 1.66, 3.45, 3.87, 1.31, 3.75) X <- matrix(c(dye,efp), length(dye), 2) colnames(X) <- c("dye","efp") BB <- BayesianBootstrap(X=X, n=100, Method=function(x,w) cov.wt(x, w, cor=TRUE)\$cor); BB #Example 5: Marginal Posterior Covariance #The following example is commented out due to package build time. #To run the following example, use the code from the examples in #the LaplaceApproximation function for the data, model specification #function, and initial values. Then perform the Laplace #Approximation as below (with CovEst="Identity" and sir=FALSE) until #convergence, set the latest initial values, then use the Bayesian #bootstrap on the data, run the Laplace Approximation again to #convergence, save the posterior modes, and repeat until S samples #of the posterior modes are collected. Finally, calculate the #parameter covariance or standard deviation. #Fit <- LaplaceApproximation(Model, Initial.Values, Data=MyData, # Iterations=1000, Method="SPG", CovEst="Identity", sir=FALSE) #Initial.Values <- as.initial.values(Fit) #S <- 100 #Number of bootstrapped sets of posterior modes (parameters) #Z <- rbind(Fit\$Summary1[,1]) #Bootstrapped parameters collected here #N <- nrow(MyData\$X) #Number of records #MyData.B <- MyData #for (s in 1:S) { # cat("\nIter:", s, "\n") # BB <- BayesianBootstrap(MyData\$y, n=N) # z <- apply(BB, 2, function(x) sample.int(N, size=1, prob=x)) # MyData.B\$y <- MyData\$y[z] # MyData.B\$X <- MyData\$X[z,] # Fit <- LaplaceApproximation(Model, Initial.Values, Data=MyData.B, # Iterations=1000, Method="SPG", CovEst="Identity", sir=FALSE) # Z <- rbind(Z, Fit\$Summary1[,1])} #cov(Z) #Bootstrapped marginal posterior covariance #sqrt(diag(cov(Z))) #Bootstrapped marginal posterior standard deviations ```