asym.var: Asymptotic covariance matrix estimation for Markov chain...

In stableGR: A Stable Gelman-Rubin Diagnostic for Markov Chain Monte Carlo

Asymptotic covariance matrix estimation for Markov chain Monte Carlo

Description

Estimates the asymptotic covariance matrix for Monte Carlo estimators, compatible with multiple chains. If a single chain is input, it calls mcmcse::mcse.multi.

Usage

asym.var(
  x,
  multivariate = TRUE,
  method = "lug",
  size = NULL,
  autoburnin = FALSE,
  adjust = TRUE
)

Arguments

x	a list of matrices, where each matrix is n \times p. 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 full matrix is returned (TRUE) or only the diagonals (FALSE)
method	the method used to compute the matrix. 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.
adjust	this argument is now obselete due to package updates.

Details

The function returns estimate of the univariate or multivariate asymptotic (co)variance of Monte Carlo estimators. If X_1, … X_n are the MCMC samples, then function returns the estimate of \lim_{n\to ∞} n Var(\bar{X}). In other words, if a Markov chain central limit holds such that, as n \to ∞

√{n}(\bar{X} - μ) \to N(0, Σ)

then the function returns an estimator of Σ from the m different chains. If multivariate == FALSE, then only the diagonal of Σ are returned.

Value

The asymptotic variance estimate (if multivariate = FALSE) or the asymptotic covariance matrix (if multivariate = TRUE) in the Markov chain central limit theorem.

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 GR diagnostic using all three chains
x <- list(chain1, chain2, chain3)
asym.var(x)

stableGR documentation built on Oct. 8, 2022, 1:05 a.m.