SeBR (version 1.1.0)

hbb: Hierarchical Bayesian bootstrap posterior sampler

Description

Compute one Monte Carlo draw from the hierarchical Bayesian bootstrap (HBB) posterior distribution of the cumulative distribution function (CDF) for each group. The common (BB) and group-specific (HBB) weights are also returned.

Usage

hbb(
  y,
  groups,
  sample_alphas = FALSE,
  shape_alphas = NULL,
  rate_alphas = NULL,
  alphas = NULL,
  M = 30
)

Value

a list with the following elements:

• Fyc: a list of functions where each entry corresponds to a group and that group-specific function can evaluate the sampled CDF at any argument(s)

• weights_y: sampled weights from the common (BB) distribution (n-dimensional)

• weights_yc: sampled weights from each of the K groups (K x n)

• alphas: the (fixed or sampled) concentration hyperparameters

Arguments

y

the data from which to infer the group-specific CDFs

groups

the group assignment for each element of y

sample_alphas

logical; if TRUE, sample the concentration hyperparameters from their marginal posterior distribution

shape_alphas

(optional) shape parameter for the Gamma prior on each alphas (if sampled)

rate_alphas

(optional) rate parameter for the Gamma prior on each alphas (if sampled)

alphas

(optional) vector of fixed concentration hyperparameters corresponding to the unique levels in groups (used when sample_alphas = FALSE)

M

a positive scaling term to set a default value of alphas when it is unspecified (alphas = NULL) and not sampled (sample_alphas = FALSE)

Details

Assuming the data y are independent with unknown, group-specific distributions, the hierarchical Bayesian bootstrap (HBB) from Oganisian et al. (<https://doi.org/10.1515/ijb-2022-0051>) is a nonparametric model for each distribution. The HBB includes hierarchical shrinkage across these groups toward a common distribution (the bb). The HBB admits direct Monte Carlo (not MCMC) sampling.

The shrinkage toward this common distribution is determined by the concentration hyperparameters alphas. Each component of alphas corresponds to one of the groups. Larger values encourage more shrinkage toward the common distribution, while smaller values allow more substantial deviations for that group.

When sample_alphas=TRUE, each component of alphas is sampled from its marginal posterior distribution, assuming independent Gamma(shape_alphas, rate_alphas) priors. This step uses a simple grid approximation to enable efficient sampling that preserves joint Monte Carlo sampling with the group-specific and common distributions. See concen_hbb for details. Note that diffuse priors on alphas tends to produce more aggressive shrinkage toward the common distribution (complete pooling). For moderate shrinkage, we use the default values shape_alphas = 30*K and rate_alphas = 1 where K is the number of groups.

When sample_alphas=FALSE, these concentration hyperparameters are fixed at user-specified values. That can be done by specifying alphas directly. Alternatively, if alphas is left unspecified (alphas = NULL), we adopt the default from Oganisian et al. which sets the cth entry to M*n/nc where M is user-specified and nc is the number of observations in group c. For further guidance on the choice of M:

• M = 0.01/K approximates separate BB's by group (no pooling);

• M between 10 and 100 gives moderate shrinkage (partial pooling); and

• M = 100*max(nc) approximates a common BB (complete pooling).

References

Oganisian et al. (<https://doi.org/10.1515/ijb-2022-0051>)

Examples

# Sample size and number of groups:
n = 500
K = 3

# Define the groups, then assign:
ugroups = paste('g', 1:K, sep='') # groups
groups = sample(ugroups, n, replace = TRUE) # assignments

# Simulate the data: iid normal, then add group-specific features
y = rnorm(n = n) # data
for(g in ugroups)
  y[groups==g] = y[groups==g] + 3*rnorm(1) # group-specific

# One draw from the HBB posterior of the CDF:
samp_hbb = hbb(y, groups)

names(samp_hbb) # items returned
Fyc = samp_hbb$Fyc # list of CDFs
class(Fyc) # this is a list
class(Fyc[[1]]) # each element is a function

c = 1 # try: vary in 1:K
Fyc[[c]](0) # some example use (for this one draw)
Fyc[[c]](c(.5, 1.2))

# Plot several draws from the HBB posterior distribution:
ys = seq(min(y), max(y), length.out=1000)
plot(ys, ys, type='n', ylim = c(0,1),
     main = 'Draws from HBB posteriors', xlab = 'y', ylab = 'F_c(y)')
for(s in 1:50){ # some draws

  # BB CDF:
  Fy = bb(y)
  lines(ys, Fy(ys), lwd=3) # plot CDF

  # HBB:
  Fyc = hbb(y, groups)$Fyc

  # Plot CDFs by group:
  for(c in 1:K) lines(ys, Fyc[[c]](ys), col=c+1, lwd=3)
}

# For reference, add the ECDFs by group:
for(c in 1:K) lines(ys, ecdf(y[groups==ugroups[c]])(ys), lty=2)

legend('bottomright', c('BB', paste('HBB:', ugroups)), col = 1:(K+1), lwd=3)