Archive for Brownian motion

scalable Langevin exact algorithm [armchair Read Paper]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on June 26, 2020 by xi'an

So, Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts presented their Read Paper with discussions on the Wednesday aft! With a well-sized if virtual audience of nearly a hundred people. Here are a few notes scribbled during the Readings. And attempts at keeping the traditional structure of the meeting alive.

In their introduction, they gave the intuition of a quasi-stationary chain as the probability to be in A at time t while still alice as π(A) x exp(-λt) for a fixed killing rate λ. The concept is quite fascinating if less straightforward than stationarity! The presentation put the stress on the available recourse to an unbiased estimator of the κ rate whose initialisation scaled as O(n) but allowed a subsampling cost reduction afterwards. With a subsampling rat connected with Bayesian asymptotics, namely on how quickly the posterior concentrates. Unfortunately, this makes the practical construction harder, since n is finite and the concentration rate is unknown (although a default guess should be √n). I wondered if the link with self-avoiding random walks was more than historical.

The initialisation of the method remains a challenge in complex environments. And hence one may wonder if and how better it does when compared with SMC. Furthermore, while the motivation for using a Brownian motion stems from the practical side, this simulation does not account for the target π. This completely blind excursion sounds worse than simulating from the prior in other settings.

One early illustration for quasi stationarity was based on an hypothetical distribution of lions and wandering (Brownian) antelopes. I found that the associated concept of soft killing was not necessarily well received by …. the antelopes!

As it happens, my friend and coauthor Natesh Pillai was the first discussant! I did no not get the details of his first bimodal example. But he addressed my earlier question about how large the running time T should be. Since the computational cost should be exploding with T. He also drew a analogy with improper posteriors as to wonder about the availability of convergence assessment.

And my friend and coauthor Nicolas Chopin was the second discussant! Starting with a request to… leave the Pima Indians (model)  alone!! But also getting into a deeper assessment of the alternative use of SMCs.

scalable Langevin exact algorithm [Read Paper]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , on June 23, 2020 by xi'an


Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts (CoI: all with whom I have strong professional and personal connections!) have a Read Paper discussion happening tomorrow [under relaxed lockdown conditions in the UK, except for the absurd quatorzine on all travelers|, but still in a virtual format] that we discussed together [from our respective homes] at Paris Dauphine. And which I already discussed on this blog when it first came out.

Here are quotes I spotted during this virtual Dauphine discussion but we did not come up with enough material to build a significant discussion, although wondering at the potential for solving the O(n) bottleneck, handling doubly intractable cases like the Ising model. And noticing the nice features of the log target being estimable by unbiased estimators. And of using control variates, for once well-justified in a non-trivial environment.

“However, in practice this simple idea is unlikely to work. We can see this most clearly with the rejection sampler, as the probability of survival will decrease exponentially with t—and thus the rejection probability will often be prohibitively large.”

“This can be viewed as a rejection sampler to simulate from μ(x,t), the distribution of the Brownian motion at time  t conditional on its surviving to time t. Any realization that has been killed is ‘rejected’ and a realization that is not killed is a draw from μ(x,t). It is easy to construct an importance sampling version of this rejection sampler.”

scalable Langevin exact algorithm

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , on October 18, 2016 by xi'an

“By employing a modification to existing naïve subsampling techniques we can obtain an algorithm which is still exact but has sub-linear iterative cost as a function of data size.”

A few weeks ago Murray Pollock, Paul Fearnhead, Adam Johansen and Gareth Roberts (all from Warwick except for Paul) arXived a paper The Scalable Langevin Exact Algorithm: Bayesian Inference for Big Data. (This was also the topic of Murray’s talk last year at JSM in Seattle.) One major advance found in the paper is the derivation of an “exact” algorithm that is sub-linear in the data size. As discussed in the introduction, the current approaches to large data problems either suffer from being approximate (like divide-and-conquer methods) or do not achieve significant reduction in the computing time, being of order O(n). The authors mention Teh and Welling (2011) sand their tochastic gradient approximation to the Langevin diffusion, when the gradient is based on a subsample. Without the Metropolis correction that would ensure an exact target but at a cost of order O(n). (Which makes the technique rather difficult to recommend.)

A novel [for me] notion at the core of this paper is the concept of quasi-stationary distribution, which is the limiting distribution of a Markov chain X[t] conditional on a Markov stopping time [being larger than t]. The approach is based on diffusions with appropriate stationary distributions like the Langevin diffusion. (Actually, as in most papers I have read and remember, the current paper only considers the Langevin diffusion.) In order to avoid the issues with unadjusted and Metropolis-adjusted Langevin schemes, a killed Brownian motion is created, which means a Brownian motion conditional of being alive till time T when the instantaneous killing rate is a given function of the chain, Φ(X[t]), related with the stationary measure of the Langevin diffusion ν. Under appropriate conditions, the density of this killed Brownian motion converges [in T] to √ν. Which immediately hints at creating a new Langevin diffusion targeting ν² instead of ν. And killing it with the proper rate, which can be done by thinning a Poisson process. Simulating the original path can be done by path-space rejection sampling, following the technique set by Gareth Roberts and co-authors more than ten years ago. Based on finite dimensional realisations of the path on [0,T]. And including the killing part can be done by importance sampling and checking that the simulated killing time is larger than the current (exponentially simulated) time.

One practical difficulty in the implementation of this neat principle is the derivation of the normalising constant, which evaluation degrades with the time horizon T. The solution adopted in the paper is through a sequential Monte Carlo method, using another discretisation of the time interval [0,T] (relying on the original one would get too costly?). As for subsampling, since the survival probability for the Brownian motion is based on an unbiased estimator, subsampling does not hurt if conducted in a random manner. Although this increases the variance on principle, the use of a control variate computed just once helps in reducing the complexity to O(1).

This is a tough paper and I have not gone through the effort of trying to implement it, but this is an original and innovative construct I would like to monitor in further details on a toy example, maybe next week while in Warwick. Or at least to discuss it with the authors.

Marc Yor

Posted in Books, Statistics, University life with tags , , , , , , on May 14, 2015 by xi'an

MarcYorMarcYor2

probabilistic numerics

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on April 27, 2015 by xi'an

sunwar2I attended an highly unusual workshop while in Warwick last week. Unusual for me, obviously. It was about probabilistic numerics, i.e., the use of probabilistic or stochastic arguments in the numerical resolution of (possibly) deterministic problems. The notion in this approach is fairly Bayesian in that it makes use to prior information or belief about the quantity of interest, e.g., a function, to construct an usually Gaussian process prior and derive both an estimator that is identical to a numerical method (e.g., Runge-Kutta or trapezoidal integration) and uncertainty or variability around this estimator. While I did not grasp much more than the classy introduction talk by Philipp Hennig, this concept sounds fairly interesting, if only because of the Bayesian connection, and I wonder if we will soon see a probability numerics section at ISBA! More seriously, placing priors on functions or functionals is a highly formal perspective (as in Bayesian non-parametrics) and it makes me wonder how much of the data (evaluation of a function at a given set of points) and how much of the prior is reflected in the output [variability]. (Obviously, one could also ask a similar question for statistical analyses!)  For instance, issues of singularity arise among those stochastic process priors.

Another question that stemmed from this talk is whether or not more efficient numerical methods can derived that way, in addition to recovering the most classical ones. Somewhat, somehow, given the idealised nature of the prior, it feels like priors could be more easily compared or ranked than in classical statistical problems. Since the aim is to figure out the value of an integral or the solution to an ODE. (Or maybe not, since again almost the same could be said about estimating a normal mean.)