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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2017/06/13/thinning-a-markov-chain-statistically/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><a rel="nofollow" href="https://xianblog.files.wordpress.com/2016/08/stanford-e1471316386177.jpg" target="_blank"><img data-orig-file="https://xianblog.files.wordpress.com/2016/08/stanford-e1471316386177.jpg?w=357&amp;h=283&amp;fit=357%2C283&amp;resize=357%2C283" data-orig-size="1433,1136" data-medium-file="https://xianblog.files.wordpress.com/2016/08/stanford-e1471316386177.jpg?w=357&amp;h=283&amp;fit=357%2C283&amp;resize=357%2C283?w=300" data-attachment-id="31307" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2016/08/stanford-e1471316386177.jpg?w=357&h=283&fit=357%2C283&resize=357%2C283" data-permalink="https://xianblog.wordpress.com/2016/08/17/mcqmc-2016-2-2/olympus-digital-camera-170/" data-large-file="https://xianblog.files.wordpress.com/2016/08/stanford-e1471316386177.jpg?w=357&amp;h=283&amp;fit=357%2C283&amp;resize=357%2C283?w=450" alt="" data-image-title="" sizes="(max-width: 357px) 100vw, 357px" data-comments-opened="1" class="wp-image-31307" data-recalc-dims="1"></a>A</strong>rt Owen has <a rel="nofollow" href="https://arxiv.org/abs/1510.07727" target="_blank">arXived</a> a new version of his <a rel="nofollow" href="https://arxiv.org/abs/1510.07727" target="_blank">thinning MCMC</a> paper, where he studies how thinning or subsampling can improve computing time in MCMC chains. I remember quite well the message set by Mark Berliner and Steve MacEachern in an early 1990’s paper that subsampling was <em>always</em> increasing the variance of the resulting estimators. We actually have this result in our <a rel="nofollow" href="http://amzn.to/2lQDmJR" target="_blank">Monte Carlo Statistical Methods</a> book. Now, there are other perspectives on this, as for instance cases when thinning can be hard-wired by simulating directly a k-step move, delaying rejection or <a rel="nofollow" href="https://xianblog.wordpress.com/2014/06/12/delayed-acceptance-with-prefetching/" target="_blank">acceptance</a>, <a rel="nofollow" href="https://xianblog.wordpress.com/2014/06/16/early-rejection-mcmc/" target="_blank">prefetching</a>, or simulating directly the accepted values as in our <a rel="nofollow" href="https://xianblog.wordpress.com/2010/06/13/vanilla-rao-blackwellisation-accepted/" target="_blank">vanilla Rao-Blackwellisation</a> approach. Here, Art considers the case when there is a cost θ of computing a transform of the simulation [when the transition cost a unit] and when those transforms are positively correlated with correlation ρ. Somewhat unsurprisingly, when θ is large enough, thinning becomes worth implementing. But requires extra computations in evaluating the correlation ρ and the cost θ, which is rarely comparable with the cost of computing the likelihood itself, a requirement for the Metropolis-Hastings or Hamiltonian Monte Carlo step(s).  Subsampling while keeping the right target (which is a hard constraint!) should thus have a much more effective impact on computing budgets.</p>
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