n.eff {stableGR}

Effective sample size

Description

For an estimator, effective sample size is the number of independent samples with the same standard error as a correlated sample. This function calculates effective sample size for a set of Markov chains using lugsail variance estimators. This also determines whether the Markov chains have converged. If they have not, this function approximates the number of samples needed.

Usage

n.eff(
  x,
  multivariate = TRUE,
  epsilon = 0.05,
  delta = NULL,
  alpha = 0.05,
  method = "lug",
  size = NULL,
  autoburnin = FALSE
)

Arguments

x

a list of matrices, where each matrix represents one Markov chain sample. Each row of the matrices represents one step of the chain. Each column of the matrices represents one variable. A list with a single matrix (chain) is allowed. Optionally, this can be an mcmclist object.

multivariate

a logical flag indicating whether the effective sample size should be calculated for multivariate chains.

epsilon

relative precision level. Values less than .10 are recommended.

delta

desired delta value - the cutoff for potential scale reduction factor.

alpha

significance level for confidence regions for the Monte Carlo estimators.

method

the method used to compute the standard error of the chains. This is one of “lug” (lugsail, the default), “bm” (batch means), “obm” (overlapping batch means), “tukey” (spectral variance method with a Tukey-Hanning window), or “bartlett” (spectral variance method with a Bartlett window).

size

options are NULL (default, which calculates an ideal batch size), character values of sqroot and cuberoot, or any numeric value between 1 and n. Size represents the batch size in “bm” (batch means) and the truncation point in “bartlett” and “tukey”. sqroot means size is floor(n^(1/2) and cuberoot means size is floor(n^(1/3)).

autoburnin

a logical flag indicating whether only the second half of the series should be used in the computation. If set to TRUE and start(x) is less than end(x)/2 then start of series will be adjusted so that only second half of series is used.

Value

n.eff

a scalar point estimate of the effective sample size.

converged

a logical indicating whether sufficient samples have been obtained.

n.target

NULL (if converged == TRUE) or a scalar estimate of the chain length required for convergence, assuming the number of chains is unchanged.

References

Vats, D. and Knudson, C. Revisiting the Gelman-Rubin Diagnostic. arXiv:1812.09384.

Vats, D. and Flegal, J. Lugsail lag windows and their application to MCMC. arXiv: 1809.04541.

Flegal, J. M. and Jones, G. L. (2010) Batch means and spectral variance estimators in Markov chain Monte Carlo. The Annals of Statistics, 38, 1034–1070.

Gelman, A and Rubin, DB (1992) Inference from iterative simulation using multiple sequences, Statistical Science, 7, 457-511.

Brooks, SP. and Gelman, A. (1998) General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7, 434-455.

Examples

library(stableGR)
set.seed(100)
p <- 2
n <- 100 # n is tiny here purely for demo purposes.
# use n much larger for real problems!


sig.mat = matrix(c(1, .8, .8, 1), ncol = 2, nrow = 2)
# Making 3 chains
chain1 <- mvn.gibbs(N = n, p = p, mu = rep(1,p), sigma = sig.mat)
chain2 <- mvn.gibbs(N = n, p = p, mu = rep(1,p), sigma = sig.mat)
chain3 <- mvn.gibbs(N = n, p = p, mu = rep(1,p), sigma = sig.mat)

# find ESS using all three chains
x <- list(chain1, chain2, chain3)
n.eff(x)