adaptive and delayed MCMC for expensive likelihoods [reply from the authors]
Thank you for the constructive criticism of our paper. Our approach uses a simple weighted average of nearest neighbours and we agree that GPs offer a useful alternative. Both methods have pros and cons, however we first note a similarity: Kriging using a GP also leads to a weighted average of values.
The two most useful pros of the GP are that, (i) by estimating the parameters of the GP one may represent the scales of variability more accurately than a simple nearest neighbour approach with weighting according to Euclidean distance, and (ii) one obtains a distribution for the uncertainty in the Kriging estimate of the log-likelihood.
Both the papers in the blog entry (as well as other recent papers which use GPs), in one way or another take advantage of the second point. However, as acknowledged in Richard Wilkinson’s paper, estimating the parameters of a GP is computationally very costly, and this estimation must be repeated as the training data set grows. Probably for this reason and because of the difficulty in identifying p(p+1)/2 kernel range parameters, Wilkinson’s paper uses a diagonal covariance structure for the kernel. We can find no description of the structure of the covariance function that is used for each statistic in the Meeds & Welling paper but this issue is difficult to avoid.
Our initial training run is used to transform the parameters so that they are approximately orthogonal with unit variance and Euclidean distance is a sensible metric. This has two consequences: (i) the KD-tree is easier to set up and use, and (ii) the nearest neighbours in a KD-tree that is approximately balanced can be found in O(log N) operations, where N is the number of training points. Both (i) and (ii) only require Euclidean distance to be a reasonable measure, not perfect, so there is no need for the training run to have “properly converged”, just for it to represent the gross relationships in the posterior and for the transformation to be 1-1. We note a parallel between our approximate standardisation using training data, and the need to estimate a symmetric matrix of distance parameters from training data to obtain a fully representative GP kernel.
The GP approach might lead to a more accurate estimate of the posterior than a nearest neighbour approach (for a fixed number of training points), but this is necessary for the algorithms in the papers mentioned above since they sample from an approximation to the posterior. As noted in the blog post the delayed-acceptance step (which also could be added to GP-based algorithms) ensures that our algorithm samples from the true posterior so accuracy is helpful for efficiency rather than essential for validity.
We have made the kd-tree C code available and put some effort into making the interface straightforward to use. Our starting point is an existing simple MCMC algorithm; as it is already evaluating the posterior (or an unbiased approximation) then why not store this and take advantage of it within the existing algorithm? We feel that our proposal offers a relatively cheap and straightforward route for this.