Archive for k-nearest neighbours

selecting summary statistics [a tale of two distances]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , on May 23, 2019 by xi'an

As Jonathan Harrison came to give a seminar in Warwick [which I could not attend], it made me aware of his paper with Ruth Baker on the selection of summaries in ABC. The setting is an ABC-SMC algorithm and it relates with Fearnhead and Prangle (2012), Barnes et al. (2012), our own random forest approach, the neural network version of Papamakarios and Murray (2016), and others. The notion here is to seek the optimal weights of different summary statistics in the tolerance distance, towards a maximization of a distance (Hellinger) between prior and ABC posterior (Wasserstein also comes to mind!). A sort of dual of the least informative prior. Estimated by a k-nearest neighbour version [based on samples from the prior and from the ABC posterior] I had never seen before. I first did not get how this k-nearest neighbour distance could be optimised in the weights since the posterior sample was already generated and (SMC) weighted, but the ABC sample can be modified by changing the [tolerance] distance weights and the resulting Hellinger distance optimised this way. (There are two distances involved, in case the above description is too murky!)

“We successfully obtain an informative unbiased posterior.”

The paper spends a significant while in demonstrating that the k-nearest neighbour estimator converges and much less on the optimisation procedure itself, which seems like a real challenge to me when facing a large number of particles and a high enough dimension (in the number of statistics). (In the examples, the size of the summary is 1 (where does the weight matter?), 32, 96, 64, with 5 10⁴, 5 10⁴, 5 10³ and…10 particles, respectively.) The authors address the issue, though, albeit briefly, by mentioning that, for the same overall computation time, the adaptive weight ABC is indeed further from the prior than a regular ABC with uniform weights [rather than weighted by the precisions]. They also argue that down-weighting some components is akin to selecting a subset of summaries, but I beg to disagree with this statement as the weights are never exactly zero, as far as I can see, hence failing to fight the curse of dimensionality. Some LASSO version could implement this feature.

adaptive and delayed MCMC for expensive likelihoods [reply from the authors]

Posted in Books, Statistics with tags , , , , , , , , on October 29, 2015 by xi'an

[Chris Sherlock, Andrew Golightly and Daniel Henderson have written a reply about my earlier comments on their arXived paper which works better as a post than as a comment:]

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.

adaptive and delayed MCMC for expensive likelihoods

Posted in Books, Statistics with tags , , , , , , , , on October 26, 2015 by xi'an

Chris Sherlock, Andrew Golightly and Daniel Henderson recently arXived a paper on a new kind of delayed acceptance.

“With simplicity in mind, we focus on a k-nearest neighbour regression model as the cheap surrogate.”

The central notion in the paper is to extrapolate from values of the likelihoods at a few points in the parameter space towards the whole space through a k-nearest neighbour estimate. While this solution is simple and relatively cheap to compute, it is unclear it is a good surrogate because it does not account for the structure of the model while depending on the choice of a distance. Recent works on Gaussian process approximations seem more relevant. See e.g. papers by Ed Meeds and Max Welling, or by Richard Wilkinson for ABC versions. Obviously, because this is a surrogate only for the first stage delayed acceptance (while the second stage is using the exact likelihood, as in our proposal), the approximation does not have to be super-tight. It should also favour the exploration of tails since (a) any proposal θ outside the current support of the chain is allocated a surrogate value that is the average of its k neighbours, hence larger than the true value in the tails, and (b) due to the delay a larger scale can be used in the random walk proposal. As the authors acknowledge, the knn method deteriorates quickly with the dimension. And computing the approximation grows with the number of MCMC iterations, given that the algorithm is adaptive and uses the exact likelihood values computed so far. Only for the first stage approximation, though, which explains “why” the delayed acceptance algorithm converges. I wondered for a short while whether this was enough to justify convergence, given that the original Metropolis-Hastings probability is just broken into two parts. Since the second stage compensates for the use of a surrogate on the first step, it should not matter in the end. However, the rejection of a proposal still depends on this approximation, i.e., differs from the original algorithm, and hence is turning the Markov chain into a non-Markovian process.

“The analysis sheds light on how computationally cheap the deterministic approximation needs to be to make its use worthwhile and on the relative importance of it matching the `location’ and curvature of the target.”

I had missed the “other” paper by some of the authors on the scaling of delayed acceptance, where they “assume that the error in the cheap deterministic approximation is a realisation of a random function” (p.3).  In which they provide an optimal scaling result for high dimensions à la Roberts et al. (1997), namely a scale of 2.38 (times the target scale) in the random walk proposal. The paper however does not describe the cheap approximation to the target or pseudo-marginal version.

A large chunk of the paper is dedicated to the construction and improvement of the KD-tree used to find the k nearest neighbours. In O(d log(n)) time. Algorithm on which I have no specific comment. Except maybe that the construction of a KD-tree in accordance with a Mahalanobis distance discussed in Section 2.1 requires that the MCMC algorithm has properly converged, which is unrealistic. And also that the construction of a balanced tree seems to require heavy calibrations.

The paper is somewhat harder to read than need be (?) because the authors cumulate the idea of delayed acceptance based on this knn approximation with the technique of pseudo-marginal Metropolis-Hastings. While there is an added value in doing so it complexifies the exposition. And leads to ungainly acronyms like adaptive “da-PsMMH”, which simply are un-readable (!).

I would suggest some material to be published as supplementary material and the overall length of the paper to be reduced. For instance, Section 4.2 is not particularly conclusive. See, e.g., Theorem 2. Or the description of the simulated models in Section 5, which is sometimes redundant.

threshold schedules for ABC-SMC

Posted in Statistics, University life with tags , , , , , , , , , on October 17, 2012 by xi'an

Daniel Silk, Sarah Filippi and Michael Stumpf just posted on arXiv a new paper on ABC-SMC. They attack therein a typical problem with SMC (and PMC, and even MCMC!) methods, namely the ability to miss (global) modes of the target due to a poor initial exploration. So, if a proposal is built on previous simulations and if  those simulations have missed an important mode, it is quite likely that this proposal will concentrate on other parts of the space and miss the important mode even more. This is also why simulated annealing and other stochastic optimisation algorithms are so fickle: a wrong parameterisation (like the temperature schedule) induces the sequence to converge to a local optimum rather than to the global optimum. Since sequential ABC is a form of simulated annealing, the decreasing tolerance (or threshold) playing the part of the temperature, it is no surprise that it suffers from this generic disease…

The method proposed in the paper aims at avoiding this difficulty by looking at sudden drops in the acceptance rate curve (as a function of the tolerance ε),

\aleph_t(\epsilon)=\int p_t(x)\mathbb{I}(\Delta(x,x^\star)\le \epsilon)\,\text{d}x,

suggesting for threshold the value of ε that maximises the second derivative of this curve. Now, before getting to (at?) the issue of turning this principle into a practical algorithm, let me linger at the motivation for it:

“To see this, one can imagine an ε-ball expanding about the true data; at first the ball only encompasses a small number of particles that were drawn from very close to the global maximum, corresponding to the low gradient at the foot of the shape. Once ε is large enough we are able to accept the relatively large number of particles sitting in the local maximum, which causes the increase in gradient.” (p.5)

Thus, the argument for looking at values of ε preceding fast increases in the acceptance rate ℵ is that we are thus avoiding flatter and lower regions of the posterior support corresponding to a local maximum. It clearly encompasses the case studied by the authors of a global highly concentrated global mode, surrounded by flatter lower modes, but it seems to me that this is not necessarily the only possible reason for changes in the shape of the acceptance probability ℵ. First, we are looking at an ABC acceptance rate, not at a genuine likelihood. Second, this acceptance rate function depends on the current (and necessarily rough) approximate to the genuine predictive, pt. Furthermore, when taking into account this rudimentary replacement of the true likelihood function, it is rather difficult to envision how it impacts the correspondence between a proximity in the data and a proximity in the parameter space. (The toy example is both illuminating and misleading, because it considers a setting where the data is a deterministic transform of the parameter.) I thus think the analysis should refer more strongly to the non-parametric literature and in particular to the k-nearest-neighbour approach recently reformulated by Biau et al.: there is no reason to push the tolerance ε all the way down to zero as this limit does not correspond to the optimal value of the tolerance. If one does not use a non-parametric determination of the “right” tolerance, the lingering question is when and why stopping the sequence of ABC simulations.

The acceptance rate function ℵ is approximated using an unscented transform. I understand neither the implementation of, nor the motivations for, this choice. Given that the function ℵ is a one-dimensional transform of the tolerance and that it actually corresponds to the cdf of the distance between the true data and the (current) pseudo-data, why couldn’t we use smooth versions of a cdf estimate based on the simulated distances, given that these distances are already computed..?  I must be missing something.

ABC as knn…

Posted in Statistics with tags , , , , on September 13, 2012 by xi'an

Gérard Biau, Frédéric Cérou, and Arnaud Guyader recently posted an arXiv paper on the foundations of ABC, entitled “New insights into Approximate Bayesian Computation“. They also submitted it to several statistics journals, with no success so far, and I find this rather surprising. Indeed, the paper analyses the ABC algorithm the way it is truly implemented (as in DIYABC for instance), i.e. with a tolerance bound ε that is determined as a quantile of the simulated distances, say the 10% or the 1% quantile. This means in particular that the interpretation of ε as a non-parametric bandwidth, while interesting and prevalent in the literature (see, e.g., Fearnhead and Prangle’s discussion paper), is only an approximation of the actual practice.

The authors of this new paper focus on the mathematical foundations of this practice, by (re)analysing ABC as a k-nearest neighbour (knn) method. Using generic knn results, they thus derive a consistency property for the ABC algorithm by imposing some constraints upon the rate of decrease of the quantile as a function of n. (The setting is restricted to the use of sufficient statistics or, equivalently, to a distance over the whole sample. The issue of summary statistics is not addressed by the paper.) The paper also contains a perfectly rigorous proof (the first one?) of the convergence of ABC when the tolerance ε goes to zero. The mean integrated square error consistency of the conditional kernel density estimate is established for a generic kernel (under usual assumptions). Further assumptions (on the target and on the kernel) allow the authors to obtain precise convergence rates (as a power of the sample size), derived from classical k-nearest neighbour regression, like

k_N \approx N^{(p+4)/(m+p+4)}

in dimensions m larger than 4…. The paper is completely theoretical and highly mathematical (with 25 pages of proofs!), which may explain why it did not meet with success with editors and/or referees, however I definitely think (an abridged version of) this work clearly deserves publication in a top statistics journal as a reference for the justification of ABC! The authors also mention future work in that direction: I would strongly suggest they consider the case of the insufficient summary statistics from this knn perspective.