As in the previous years, I am back in Oxford (England) for my short Bayesian Statistics course in the joint Oxford-Warwick PhD programme, OxWaSP. For some unclear reason, presumably related to the Internet connection from Oxford, I have not been able to upload my slides to Slideshare, so here the [99.9% identical] older version:
Archive for University of Warwick
Lawrence Murray, Sumeet Singh, Pierre Jacob, and Anthony Lee (Warwick) recently arXived a paper on Anytime Monte Carlo. (The earlier post on this topic is no coincidence, as Lawrence had told me about this problem when he visited Paris last Spring. Including a forced extension when his passport got stolen.) The difficulty with anytime algorithms for MCMC is the lack of exchangeability of the MCMC sequence (except for formal settings where regeneration can be used).
When accounting for duration of computation between steps of an MCMC generation, the Markov chain turns into a Markov jump process, whose stationary distribution α is biased by the average delivery time. Unless it is constant. The authors manage this difficulty by interlocking the original chain with a secondary chain so that even- and odd-index chains are independent. The secondary chain is then discarded. This provides a way to run an anytime MCMC. The principle can be extended to K+1 chains, run one after the other, since only one of those chains need be discarded. It also applies to SMC and SMC². The appeal of anytime simulation in this particle setting is that resampling is no longer a bottleneck. Hence easily distributed among processors. One aspect I do not fully understand is how the computing budget is handled, since allocating the same real time to each iteration of SMC seems to envision each target in the sequence as requiring the same amount of time. (An interesting side remark made in this paper is the lack of exchangeability resulting from elaborate resampling mechanisms, lack I had not thought of before.)
Today, I alas missed a seminar at BiPS on the Zig-Zag (sub-)sampler of Joris Bierkens, Paul Fearnhead and Gareth Roberts, presented here in Paris by James Ridgway. Fortunately for me, I had some discussions with Murray Pollock in Warwick and then again with Changye Wu in Dauphine that shed some light on this complex but highly innovative approach to simulating in Big Data settings thanks to a correct subsampling mechanism.
The zig-zag process runs a continuous process made of segments that turn from one diagonal to the next at random times driven by a generator connected with the components of the gradient of the target log-density. Plus a symmetric term. Provided those random times can be generated, this process is truly available and associated with the right target distribution. When the components of the parameter are independent (an unlikely setting), those random times can be associated with an inhomogeneous Poisson process. In the general case, one needs to bound the gradients by more manageable functions that create a Poisson process that can later be thinned. Next, one needs to simulate the process for the upper bound, a task that seems hard to achieve apart from linear and piecewise constant upper bounds. The process has a bit of a slice sampling taste, except that it cannot be used as a slice sampler but requires continuous time integration, given that the length of each segment matters. (Or maybe random time subsampling?)
A highly innovative part of the paper concentrates on Big Data likelihoods and on the possibility to subsample properly and exactly the original dataset. The authors propose Zig-Zag with subsampling by turning the gradients into random parts of the gradients. While remaining unbiased. There may be a cost associated with this gain of one to n, namely that the upper bounds may turn larger as they handle all elements in the likelihood at once, hence become (even) less efficient. (I am more uncertain about the case of the control variates, as it relies on a Lipschitz assumption.) While I still miss an easy way to implement the approach in a specific model, I remain hopeful for this new approach to make a major dent in the current methodologies!
Possibly the last post on random number generation by Kinderman and Monahan’s (1977) ratio-of-uniform method. After fiddling with the Gamma(a,1) distribution when a<1 for a while, I indeed figured out a way to produce a bounded set with this method: considering an arbitrary cdf Φ with corresponding pdf φ, the uniform distribution on the set Λ of (u,v)’s in R⁺xX such that
induces the distribution with density proportional to ƒ on φοΦ⁻¹(U)V. This set Λ has a boundary that is parameterised as
u=Φοƒ(x), v=1/φοƒ(x), x∈Χ
which remains bounded in u since Φ is a cdf and in v if φ has fat enough tails. At both 0 and ∞. When ƒ is the Gamma(a,1) density this can be achieved if φ behaves like log(x)² near zero and like a inverse power at infinity. Without getting into all the gory details, closed form density φ and cdf Φ can be constructed for all a’s, as shown for a=½ by the boundaries in u and v (yellow) below
At this stage, I remain uncertain of the relevance of such derivations, if only because the set A thus derived is ill-suited for uniform draws proposed on the enclosing square box. And also because a Gamma(a,1) simulation can rather simply be derived from a Gamma(a+1,1) simulation. But, who knows?!, there may be alternative usages of this representation, such as innovative slice samplers. Which means the ratio-of-uniform method may reappear on the ‘Og one of those days…
Seven academic positions in Statistics are currently opening at the University of Warwick! They correspond to the following levels, all permanent but for the Harrison Early-Career Assistant Professorship:
• Harrison Early-Career Assistant Professor of Statistics (3 years)
• Assistant Professor of Statistics
• Senior Teaching Fellow in Statistics
• Assistant or Associate Professor of Financial Mathematics
• Assistant or Associate Professor of Statistics (two positions)
• Full Professor of Statistics
Applicants are sought with expertise in Statistics (in the wide sense, including both applied and methodological statistics, probability, probabilistic operational research and mathematical finance together with interdisciplinary topics involving one or more of these areas). Applicants for senior positions should have an excellent publication record and ability to secure research funding. Applicants for more junior positions should show exceptional promise to become leading academics. [More details.]
Informal enquires should be addressed to any of Professors Mark Steel, David Hobson, Gareth Roberts, Wilfrid Kendall, or to any other senior member of the Department. Including me. The closing date is 3 January 2017 for all positions, except for the Full Professor position which is set to 10 January 2017.
As a completely objective observer (!), I can state that this is a fantastic department, with a strong sense of community and support among the faculty, and a tremendously diverse array of research activities and topics. And thus encourage anyone interested in joining to apply for some of those positions!
A week ago, Chris Sherlock, Alexandre Thiery and Anthony Lee (Warwick) arXived a short paper where they show that it is most efficient to use only one random substitute to the likelihood when considering a pseudo-marginal algorithm. Thus generalising an earlier paper of Luke Bornn and co-authors I commented earlier. A neat side result in the paper is that the acceptance probability for m replicas [in the pseudo-marginal approximation] is at most m/s the acceptance probability for s replicas when s<m. The main result is as follows:
There is a (superficial?) connection with the fact that when running Metropolis-within-Gibbs there is no advantage in doing more than one single Metropolis iteration, as the goal is to converge to the joint posterior, rather than approximating better the full conditional…