  It is common to subsample Markov chain output to reduce the storage burden.
Geyer (1992) shows that discarding $k-1$ out of every $k$ observations will not
improve statistical efficiency, as quantified through variance in a given
computational budget. That observation is often taken to mean that thinning
MCMC output cannot improve statistical efficiency. Here we suppose that it
costs one unit of time to advance a Markov chain and then $\theta>0$ units of
time to compute a sampled quantity of interest. For a thinned process, that
cost $\theta$ is incurred less often, so it can be advanced through more
stages. Here we provide examples to show that thinning will improve statistical
efficiency if $\theta$ is large and the sample autocorrelations decay slowly
enough. If the lag $\ell\ge1$ autocorrelations of a scalar measurement satisfy
$\rho_\ell\ge\rho_{\ell+1}\ge0$, then there is always a $\theta<\infty$ at
which thinning becomes more efficient for averages of that scalar. Many sample
autocorrelation functions resemble first order AR(1) processes with $\rho_\ell
=\rho^{|\ell|}$ for some $-1<\rho<1$. For an AR(1) process it is possible to
compute the most efficient subsampling frequency $k$. The optimal $k$ grows
rapidly as $\rho$ increases towards $1$. The resulting efficiency gain depends
primarily on $\theta$, not $\rho$. Taking $k=1$ (no thinning) is optimal when
$\rho\le0$. For $\rho>0$ it is optimal if and only if $\theta \le
(1-\rho)^2/(2\rho)$. This efficiency gain never exceeds $1+\theta$. This paper
also gives efficiency bounds for autocorrelations bounded between those of two
AR(1) processes.
