This function may be used to estimate the effective sample size (ESS) (not to be confused with Elliptical Slice Sampling) of a continuous target distribution, where the sample size is reduced by autocorrelation. ESS is a measure of how well each continuous chain is mixing.
ESS is a univariate function that is often applied to each continuous, marginal posterior distribution. A multivariate form is not included. By chance alone due to multiple independent tests, 5% of the continuous parameters may indicate that ESS is below a user threshold of acceptability, such as 100, even when above the threshold. Assessing convergence is difficult.
ESS(x)This required argument is a vector or matrix of posterior samples.
A vector is returned, and each element is the effective sample size
(ESS) for a corresponding column of x, after autocorrelation has
been taken into account.
Effective Sample Size (ESS) was recommended by Radford Neal in the
panel discussion of Kass et al. (1998). When a continuous, marginal
posterior distribution is sampled with a Markov chain Monte Carlo
(MCMC) algorithm, there is usually autocorrelation present in the
samples. More autocorrelation is associated with less posterior
sampled information, because the information in the samples is
autocorrelated, or put another way, successive samples are not
independent from earlier samples. This reduces the effective sample
size of, and precision in representing, the continuous, marginal
posterior distribution. ESS is one of the criteria in the
Consort function, where stopping the MCMC updates is
not recommended until ESS \(\ge 100\). Although the need
for precision of each modeler differs with each model, it is often
a good goal to obtain ESS \(= 1000\).
ESS is related to the integrated autocorrelation time (see
IAT for more information).
ESS is usually defined as
$$\mathrm{ESS}(\theta) = \frac{S}{1 + 2 \sum^{\infty}_{k=1} \rho_k (\theta)},$$
where \(S\) is the number of posterior samples,
\(\rho_k\) is the autocorrelation at lag \(k\), and
\(\theta\) is the vector of marginal posterior samples. The
infinite sum is often truncated at lag \(k\) when
\(\rho_k (\theta) < 0.05\). Just as with the
effectiveSize function in the coda package, the
AIC argument in the ar function is used to estimate the
order.
ESS is a measure of how well each continuous chain is mixing, and a
continuous chain mixes better when in the target distribution. This
does not imply that a poorly mixing chain still searching for its
target distribution will suddenly mix well after finding it, though
mixing should improve. A poorly mixing continuous chain does not
necessarily indicate problems. A smaller ESS is often due to
correlated parameters, and is commonly found with scale
parameters. Posterior correlation may be obtained from the
PosteriorChecks function, and plotted with the
plotMatrix function. Common remedies for poor mixing
include re-parameterizing the model or trying a different MCMC
algorithm that better handles correlated parameters. Slow mixing is
indicative of an inefficiency in which a continuous chain takes longer
to find its target distribution, and once found, takes longer to
explore it. Therefore, slow mixing results in a longer required
run-time to find and adequately represent the continuous target
distribution, and increases the chance that the user may make
inferences from a less than adequate representation of the continuous
target distribution.
There are many methods of re-parameterization to improve mixing. It
is helpful when predictors are centered and scaled, such as with the
CenterScale function. Parameters for predictors are
often assigned prior distributions that are independent per parameter,
in which case an exchangeable prior distribution or a multivariate
prior distribution may help. If a parameter with poor mixing is
bounded with the interval function, then
transforming it to the real line (such as with a log transformation
for a scale parameter) is often helpful, since constraining a
parameter to an interval often reduces ESS. Another method is to
re-parameterize so that one or more latent variables represent the
process that results in slow mixing. Such re-parameterization uses
data augmentation.
This is numerically the same as the effectiveSize function in
the coda package, but programmed to accept a simple vector or
matrix so it does not require an mcmc or mcmc.list
object, and the result is bound to be less than or equal to the
original number of samples.
Kass, R.E., Carlin, B.P., Gelman, A., and Neal, R. (1998). "Markov Chain Monte Carlo in Practice: A Roundtable Discussion". The American Statistician, 52, p. 93--100.
CenterScale,
Consort,
IAT,
interval,
LaplacesDemon,
plotMatrix, and
PosteriorChecks.