Archive for Poisson process

sampling and imbalanced

Posted in Statistics with tags , , , , , on June 21, 2019 by xi'an

Deborshee Sen, Matthias Sachs, Jianfeng Lu and David Dunson have recently arXived a sub-sampling paper for  classification (logistic) models where some covariates or some responses are imbalanced. With a PDMP, namely zig-zag, used towards preserving the correct invariant distribution (as already mentioned in an earlier post on the zig-zag zampler and in a recent Annals paper by Joris Bierkens, Paul Fearnhead, and Gareth Roberts (Warwick)). The current paper is thus an improvement on the above. Using (non-uniform) importance sub-sampling across observations and simpler upper bounds for the Poisson process. A rather practical form of Poisson thinning. And proposing unbiased estimates of the sub-sample log-posterior as well as stratified sub-sampling.

I idly wondered if the zig-zag sampler could itself be improved by not switching the bouncing directions at random since directions associated with almost certainly null coefficients should be neglected as much as possible, but the intensity functions associated with the directions do incorporate this feature. Except for requiring computation of the intensities for all directions. This is especially true when facing many covariates.

Thinking of the logistic regression model itself, it is sort of frustrating that something so close to an exponential family causes so many headaches! Formally, it is an exponential family but the normalising constant is rather unwieldy, especially when there are many observations and many covariates. The Polya-Gamma completion is a way around, but it proves highly costly when the dimension is large…

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.

The answer is e, what was the question?!

Posted in Books, R, Statistics with tags , , , , , on February 12, 2016 by xi'an

Sceaux, June 05, 2011A rather exotic question on X validated: since π can be approximated by random sampling over a unit square, is there an equivalent for approximating e? This is an interesting question, as, indeed, why not focus on e rather than π after all?! But very quickly the very artificiality of the problem comes back to hit one in one’s face… With no restriction, it is straightforward to think of a Monte Carlo average that converges to e as the number of simulations grows to infinity. However, such methods like Poisson and normal simulations require some complex functions like sine, cosine, or exponential… But then someone came up with a connection to the great Russian probabilist Gnedenko, who gave as an exercise that the average number of uniforms one needs to add to exceed 1 is exactly e, because it writes as

\sum_{n=0}^\infty\frac{1}{n!}=e

(The result was later detailed in the American Statistician as an introductory simulation exercise akin to Buffon’s needle.) This is a brilliant solution as it does not involve anything but a standard uniform generator. I do not think it relates in any close way to the generation from a Poisson process with parameter λ=1 where the probability to exceed one in one step is e⁻¹, hence deriving  a Geometric variable from this process leads to an unbiased estimator of e as well. As an aside, W. Huber proposed the following elegantly concise line of R code to implement an approximation of e:

1/mean(n*diff(sort(runif(n+1))) > 1)

Hard to beat, isn’t it?! (Although it is more exactly a Monte Carlo approximation of

\left(1-\frac{1}{n}\right)^n

which adds a further level of approximation to the solution….)

an extension of nested sampling

Posted in Books, Statistics, University life with tags , , , , , , , on December 16, 2014 by xi'an

I was reading [in the Paris métro] Hastings-Metropolis algorithm on Markov chains for small-probability estimation, arXived a few weeks ago by François Bachoc, Lionel Lenôtre, and Achref Bachouch, when I came upon their first algorithm that reminded me much of nested sampling: the following was proposed by Guyader et al. in 2011,

To approximate a tail probability P(H(X)>h),

  • start from an iid sample of size N from the reference distribution;
  • at each iteration m, select the point x with the smallest H(x)=ξ and replace it with a new point y simulated under the constraint H(y)≥ξ;
  • stop when all points in the sample are such that H(X)>h;
  • take

\left(1-\dfrac{1}{N}\right)^{m-1}

as the unbiased estimator of P(H(X)>h).

Hence, except for the stopping rule, this is the same implementation as nested sampling. Furthermore, Guyader et al. (2011) also take advantage of the bested sampling fact that, if direct simulation under the constraint H(y)≥ξ is infeasible, simulating via one single step of a Metropolis-Hastings algorithm is as valid as direct simulation. (I could not access the paper, but the reference list of Guyader et al. (2011) includes both original papers by John Skilling, so the connection must be made in the paper.) What I find most interesting in this algorithm is that it even achieves unbiasedness (even in the MCMC case!).