## Introduction

Markov chain Monte Carlo (MCMC) is a technique (or more correctly, a family of techniques) for sampling probability distributions. Typical applications are in Bayesian modelling, the target distributions being posterior distributions of unknown parameters, or predictive distributions for unobserved phenomena. MCMC is becoming commonplace as a tool for fitting ecological models. The first applications of MCMC methods in publications of American and British ecological societies were in a paper published by the British Ecological Society (BES) in 2001 (Groombridge *et al.* 2001) and in five papers published by the Ecological Society of America (ESA) in 2002 (Gross, Craig, & Hutchison 2002; Link & Sauer 2002; Mac Nally & Fleishman 2002; O’Hara *et al.* 2002; Sauer & Link 2002). Since then, the use of MCMC in journals of these societies has increased rapidly. Summarising over three publications of the Ecological Society of America (*ESA*: *Ecology*, *Ecological Applications* and *Ecological Monographs*) and five publications of the British Ecological Society (*BES: J. of Ecology, J. of Applied Ecology*, *Functional Ecology*, *J. of Animal Ecology* and *Methods in Ecology and Evolution*), the numbers of publications using MCMC were 1, 6, 12, 10, 14, 21, 13, 28, 49 and 45, for years 2001–2010.

The appeal of MCMC is that it is almost always relatively easy to implement, even when the target distributions are complicated and conventional simulation techniques are impossible. The difference between MCMC and traditional simulation methods is that MCMC produces a dependent sequence *–* a Markov chain *–* of values, rather than a sequence of independent draws. The Markov chain sample is summarised just like a conventional independent sample; sample features (e.g. mean, variance and percentiles) are used to approximate corresponding features of the target distribution. The disadvantage of MCMC is that these approximations are typically less precise than would be obtained from an independent sample of the same size.

Many practitioners routinely thin their chains *–* that is, they discard all but every *k*th observation *–* with the goal of reducing autocorrelation. Among 76 *Ecology* papers published between 2002 and 2010, 15 mentioned MCMC, but did not apply it; eight used MCMC, but provided no details on the actual implementation. Twenty-one of the remaining 53 (40%) reported thinning; among these, the median rate of thinning was to select every 40th value (‘×40’ thinning). Five studies reported thinning rates of ×750 or higher, and the highest rate was ×10^{5}. Among 73 papers published in five journals of the BES, 27 mentioned MCMC but either did not apply it or used packaged software developed for genetic analyses that offered limited user-control over the implementation of MCMC. A further nine publications applied MCMC methods but provided no details on its implementation. Fifteen of the remaining 37 (41%) reported thinning of chains. The median thinning rate among these studies was ×29, and the highest was ×1000.

Our purpose in writing this note is to discourage the practice of thinning, which is usually unnecessary, and always inefficient. Our observation is not a new one: MacEachern & Berliner (1994) provide ‘a justification of the ban [on] subsampling’ MCMC output; see also Geyer (1992). We are not suggesting or promoting a ban on the practice; there are circumstances (discussed later) where thinning is reasonable. In these cases, we encourage the practitioner to be explicit in his or her reasoning for sacrificing one sort of efficiency for another. However, for approximation of simple features of the target distribution (e.g. means, variances and percentiles), thinning is neither necessary nor desirable; results based on unthinned chains are more precise.

We write this note assuming readers have some acquaintance with MCMC methods; for more details on fundamentals, we refer readers to Link *et al.* (2002) or to texts by Gelman *et al.* (2004) and Link & Barker (2010). Because our emphasis is on the practice of thinning chains, we assume that MCMC output follows from appropriate starting values and adequate burnin to allow evaluation as stationary chains.