R: Monte Carlo Standard Error

MCSE {LaplacesDemon} | R Documentation

## Monte Carlo Standard Error

### Description

Monte Carlo Standard Error (MCSE) is an estimate of the inaccuracy of Monte Carlo samples, usually regarding the expectation of posterior samples, E(theta), from Monte Carlo or Markov chain Monte Carlo (MCMC) algorithms, such as with the `LaplacesDemon` or `LaplacesDemon.hpc` functions. MCSE approaches zero as the number of independent posterior samples approaches infinity. MCSE is essentially a standard deviation around the posterior mean of the samples, E(theta), due to uncertainty associated with using an MCMC algorithm, or Monte Carlo methods in general.

The acceptable size of the MCSE depends on the acceptable uncertainty associated around the marginal posterior mean, E(theta), and the goal of inference. It has been argued that MCSE is generally unimportant when the goal of inference is theta rather than E(theta) (Gelman et al., 2004, p. 277), and that a sufficient `ESS` is more important. Others perceive MCSE to be a vital part of reporting any Bayesian model, and as a stopping rule (Flegal et al., 2008).

In `LaplacesDemon`, MCSE is part of the posterior summaries because it is easy to estimate, and Laplace's Demon prefers to continue updating until each MCSE is less than 6.27% of its associated marginal posterior standard deviation (for more information on this stopping rule, see the `Consort` function), since MCSE has been demonstrated to be an excellent stopping rule.

Acceptable error may be specified, if known, in the `MCSS` (Monte Carlo Sample Size) function to estimate the required number of posterior samples.

`MCSE` is a univariate function that is often applied to each marginal posterior distribution. A multivariate form is not included. By chance alone due to multiple independent tests, 5% of the parameters should indicate unacceptable MSCEs, even when acceptable. Assessing convergence is difficult.

### Usage

    MCSE(x, method="IMPS", batch.size="sqrt", warn=FALSE)
    MCSS(x, a)

### Arguments

`x` | This is a vector of posterior samples for which MCSE or MCSS will be estimated.
`a` | This is a scalar argument of acceptable error for the mean of `x`, and `a` must be positive. As acceptable error decreases, the required number of samples increases.
`method` | This is an optional argument for the method of MCSE estimation, and defaults to Geyer's `"IMPS"` method. Optional methods include `"sample.variance"` and `"batch.mean"`. Note that `"batch.mean"` is recommended only when the number of posterior samples is at least 1,000.
`batch.size` | This is an optional argument that corresponds only with `method="batch.means"`, and determines either the size of the batches (accepting a numerical argument) or the method of creating the size of batches, which is either `"sqrt"` or `"cuberoot"`, and refers to the length of `x`. The default argument is `"sqrt"`.
`warn` | Logical. If `warn=TRUE`, then a warning is provided with `method="batch.means"` whenever posterior sample size is less than 1,000, or a warning is produced when more autcovariance is recommended with `method="IMPS"`.

### Details

The default method for estimating MCSE is Geyer's Initial Monotone Positive Sequence (IMPS) estimator (Geyer, 1992), which takes the asymptotic variance into account and is time-series based. This method goes by other names, such as Initial Positive Sequence (IPS).

The simplest method for estimating MCSE is to modify the formula for standard error, sigma(x) / sqrt(N), to account for non-independence in the sequence x of posterior samples. Non-independence is estimated with the `ESS` function for Effective Sample Size (see the `ESS` function for more details), where m = ESS(x), and MCSE is sigma(x) / sqrt(M). Although this is the fastest and easiest method of estimation, it does not incorporate an estimate of the asymptotic variance of x.

The batch means method (Jones et al., 2006; Flegal et al., 2008) separates elements of x into batches and estimates MCSE as a function of multiple batches. This method is excellent, but is not recommended when the number of posterior samples is less than 1,000. These journal articles also assert that MCSE is a better stopping rule than MCMC convergence diagnostics.

The `MCSS` function estimates the required number of posterior samples, given the user-specified acceptable error, posterior samples `x`, and the observed variance (rather than asymptotic variance). Due to the observed variance, this is a rough estimate.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### References

Flegal, J.M., Haran, M., and Jones, G.L. (2008). "Markov chain Monte Carlo: Can We Trust the Third Significant Figure?". Statistical Science, 23, p. 250–260.

Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2004). "Bayesian Data Analysis, Texts in Statistical Science, 2nd ed.". Chapman and Hall, London.

Geyer, C.J. (1992). "Practical Markov Chain Monte Carlo". Statistical Science, 7, 4, p. 473–483.

Jones, G.L., Haran, M., Caffo, B.S., and Neath, R. (2006). "Fixed-Width Output Analysis for Markov chain Monte Carlo". Journal of the American Statistical Association, 101(1), p. 1537–1547.

### See Also

`Consort`, `ESS`, `LaplacesDemon`, and `LaplacesDemon.hpc`.

### Examples

    library(LaplacesDemon)
    x <- rnorm(1000)
    MCSE(x)
    MCSE(x, method="batch.means")
    MCSE(x, method="sample.variance")
    MCSS(x, a=0.01)

[Package LaplacesDemon version 16.1.1 Index]