Statistically efficient thinning of a Markov chain sampler

Abstract

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 $θ>0$ units of time to compute a sampled quantity of interest. For a thinned process, that cost $θ$ 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 $θ$ is large and the sample autocorrelations decay slowly enough. If the lag $\ell\ge1$ autocorrelations of a scalar measurement satisfy $ρ_\ell\geρ_{\ell+1}\ge0$, then there is always a $θ<\infty$ at which thinning becomes more efficient for averages of that scalar. Many sample autocorrelation functions resemble first order AR(1) processes with $ρ_\ell =ρ^{|\ell|}$ for some $-1<ρ<1$. For an AR(1) process it is possible to compute the most efficient subsampling frequency $k$. The optimal $k$ grows rapidly as $ρ$ increases towards $1$. The resulting efficiency gain depends primarily on $θ$, not $ρ$. Taking $k=1$ (no thinning) is optimal when $ρ\le0$. For $ρ>0$ it is optimal if and only if $θ\le (1-ρ)^2/(2ρ)$. This efficiency gain never exceeds $1+θ$. This paper also gives efficiency bounds for autocorrelations bounded between those of two AR(1) processes.