## 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.

## importance sampling by kernel smoothing [experiment]

Posted in Books, R, Statistics with tags , , , , , , on October 13, 2016 by xi'an

Following my earlier post on Delyon and Portier’s proposal to replacing the true importance distribution ƒ with a leave-one-out (!) kernel estimate in the importance sampling estimator, I ran a simple one-dimensional experiment to compare the performances of the traditional method with this alternative. The true distribution is a N(0,½) with an importance proposal a N(0,1) distribution, the target is the function h(x)=x⁶ [1-0.9 sin(3x)], n=2643 is the number of simulations, and the density is estimated via the call to the default density() R function. The first three boxes are for the regular importance sampler, and the kernel and the corrected kernel versions of Delyon and Portier, while the second set of three considers the self-normalised alternatives. In all kernel versions, the variability is indeed much lower than with importance sampling, but the bias is persistent, with no clear correction brought by the first order proposal in the paper, while those induce a significant increase in computing time:

> benchmark(
+ for (t in 1:100){
+   x=sort(rnorm(N));fx=dnorm(x)
+  imp1=dnorm(x,sd=.5)/fx})

replicas elapsed relative user.child sys.child
1        100     7.948    7.94       0.012
> benchmark(
+ for (t in 1:100){
+   x=sort(rnorm(N));hatf=density(x)
+   hatfx=approx(hatf$x,hatf$y, x)$y + imp2=dnorm(x,sd=.5)/hatfx}) replicas elapsed relative user.child sys.child 1 100 19.272 18.473 0.94 > benchmark( + for (t in 1:100){ + x=sort(rnorm(N));hatf=density(x) + hatfx=approx(hatf$x,hatf$y, x)$y
+   bw=hatf\$bw
+   for (i in 1:N) Kx[i]=1-sum((dnorm(x[i],
+     mean=x[-i],sd=bw)-hatfx[i])^2)/NmoNmt/hatfx[i]^2
+   imp3=dnorm(x,sd=.5)*Kx/hatfx})

replicas elapsed relative user.child sys.child
1        100     11378.38  7610.037  17.239


which follows from the O(n) cost in deriving the kernel estimate for all observations (and I did not even use the leave-one-out option…) The R computation of the variance is certainly not optimal, far from it, but those enormous values give an indication of the added cost of the step, which does not even seem productive in terms of variance reduction… [Warning: the comparison is only done over one model and one target integrand, thus does not pretend at generality!]

## importance sampling by kernel smoothing

Posted in Books, Statistics with tags , , , , , on September 27, 2016 by xi'an

As noted in an earlier post, Bernard Delyon and François Portier have recently published a paper in Bernoulli about improving the speed of convergence of an importance sampling estimator of

∫ φ(x) dx

when replacing the true importance distribution ƒ with a leave-one-out (!) kernel estimate in the importance sampling estimator… They also consider a debiased version that converges even faster at the rate

$n h_n^{d/2}$

where n is the sample size, h the bandwidth and d the dimension. There is however a caveat, namely a collection of restrictive assumptions on the components of this new estimator:

1. the integrand φ has a compact support, is bounded, and satisfies some Hölder-type regularity condition;
2. the importance distribution ƒ is upper and lower bounded, its r-th order derivatives are upper bounded;
3. the kernel K is order r, with exponential tails, and symmetric;
4. the leave-one-out correction for bias has a cost O(n²) compared with O(n) cost of the regular Monte-Carlo estimator;
5. the bandwidth h in the kernel estimator has a rate in n linked with the dimension d and the regularity indices of ƒ and φ

and this bandwidth needs to be evaluated as well. In the paper the authors rely on a control variate for which the integral is known, but which “looks like φ”, a strong requirement in appearance only since this new function is the convolution of φ with a kernel estimate of ƒ which expectation is the original importance estimate of the integral. This sounds convoluted but this is a generic control variate nonetheless! But this is also a costly step. Because of the kernel estimation aspect, the method deteriorates with the dimension of the variate x. However, since φ(x) is a real number, I wonder if running the non-parametric density estimate directly on the sample of φ(x)’s would lead to an improved estimator…

## merging MCMC subposteriors

Posted in Books, Statistics, University life with tags , , , , , , , on June 8, 2016 by xi'an

Christopher Nemeth and Chris Sherlock arXived a paper yesterday about an approach to distributed MCMC sampling via Gaussian processes. As in several other papers commented on the ‘Og, the issue is to merge MCMC samples from sub-posteriors into a sample or any sort of approximation of the complete (product) posterior. I am quite sympathetic to the approach adopted in this paper, namely to use a log-Gaussian process representation of each sub-posterior and then to replace each sub-posterior with its log-Gaussian process posterior expectation in an MCMC or importance scheme. And to assess its variability through the posterior variance of the sum of log-Gaussian processes. As pointed out by the authors the closed form representation of the posterior mean of the log-posterior is invaluable as it allows for an HMC implementation. And importance solutions as well. The probabilistic numerics behind this perspective are also highly relevant.

A few arguable (?) points:

1. The method often relies on importance sampling and hence on the choice of an importance function that is most likely influential but delicate to calibrate in complex settings as I presume the Gaussian estimates are not useful in this regard;
2. Using Monte Carlo to approximate the value of the approximate density at a given parameter value (by simulating from the posterior distribution) is natural but is it that efficient?
3. It could be that, by treating all sub-posterior samples as noisy versions of the same (true) posterior, a more accurate approximation of this posterior could be constructed;
4. The method relies on the exponentiation of a posterior expectation or simulation. As of yesterday, I am somehow wary of log-normal expectations!
5. If the purpose of the exercise is to approximate univariate integrals, it would seem more profitable to use the Gaussian processes at the univariate level;
6. The way the normalising missing constants and the duplicate simulations are processed (or not) could deserve further exploration;
7. Computing costs are in fine unclear when compared with the other methods in the toolbox.

## inefficiency of data augmentation for large samples

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , on May 31, 2016 by xi'an

On Monday, James Johndrow, Aaron Smith, Natesh Pillai, and David Dunson arXived a paper on the diminishing benefits of using data augmentation for large and highly imbalanced categorical data. They reconsider the data augmentation scheme of Tanner and Wong (1987), surprisingly not mentioned, used in the first occurrences of the Gibbs sampler like Albert and Chib’s (1993) or our mixture estimation paper with Jean Diebolt (1990). The central difficulty with data augmentation is that the distribution to be simulated operates on a space that is of order O(n), even when the original distribution covers a single parameter. As illustrated by the coalescent in population genetics (and the subsequent intrusion of the ABC methodology), there are well-known cases when the completion is near to impossible and clearly inefficient (as again illustrated by the failure of importance sampling strategies on the coalescent). The paper provides spectral gaps for the logistic and probit regression completions, which are of order a power of log(n) divided by √n, when all observations are equal to one. In a somewhat related paper with Jim Hobert and Vivek Roy, we studied the spectral gap for mixtures with a small number of observations: I wonder at the existence of a similar result in this setting, when all observations stem from one component of the mixture, when all observations are one. The result in this paper is theoretically appealing, the more because the posteriors associated with such models are highly regular and very close to Gaussian (and hence not that challenging as argued by Chopin and Ridgway). And because the data augmentation algorithm is uniformly ergodic in this setting (as we established with Jean Diebolt  and later explored with Richard Tweedie). As demonstrated in the  experiment produced in the paper, when comparing with HMC and Metropolis-Hastings (same computing times?), which produce much higher effective sample sizes.

## likelihood inflating sampling algorithm

Posted in Books, Statistics, University life with tags , , , , , , , , on May 24, 2016 by xi'an

My friends from Toronto Radu Craiu and Jeff Rosenthal have arXived a paper along with Reihaneh Entezari on MCMC scaling for large datasets, in the spirit of Scott et al.’s (2013) consensus Monte Carlo. They devised an likelihood inflated algorithm that brings a novel perspective to the problem of large datasets. This question relates to earlier approaches like consensus Monte Carlo, but also kernel and Weierstrass subsampling, already discussed on this blog, as well as current research I am conducting with my PhD student Changye Wu. The approach by Entezari et al. is somewhat similar to consensus Monte Carlo and the other solutions in that they consider an inflated (i.e., one taken to the right power) likelihood based on a subsample, with the full sample being recovered by importance sampling. Somewhat unsurprisingly this approach leads to a less dispersed estimator than consensus Monte Carlo (Theorem 1). And the paper only draws a comparison with that sub-sampling method, rather than covering other approaches to the problem, maybe because this is the most natural connection, one approach being the k-th power of the other approach.

“…we will show that [importance sampling] is unnecessary in many instances…” (p.6)

An obvious question that stems from the approach is the call for importance sampling, since the numerator of the importance sampler involves the full likelihood which is unavailable in most instances when sub-sampled MCMC is required. I may have missed the part of the paper where the above statement is discussed, but the only realistic example discussed therein is the Bayesian regression tree (BART) of Chipman et al. (1998). Which indeed constitutes a challenging if one-dimensional example, but also one that requires delicate tuning that leads to cancelling importance weights but which may prove delicate to extrapolate to other models.

## Monte Carlo methods for Potts models

Posted in pictures, Statistics, University life with tags , , , , on March 10, 2016 by xi'an

There will be a seminar talk by Mehdi Molkaraie (Pompeu Fabra) next week at Institut Henri Poincaré (IHP), Paris, on his paper with Vincent Gomez.

We consider the problem of estimating the partition function of the ferromagnetic q-state Potts model. We propose an importance sampling algorithm in the dual of the normal factor graph representing the model. The algorithm can efficiently compute an estimate of the partition function when the coupling parameters of the model are strong (corresponding to models at low temperature) or when the model contains a mixture of strong and weak couplings. We show that, in this setting, the proposed algorithm significantly outperforms the state of the art methods.

The talk is at 14:30, March 17. It is part of a trimester program on information and computation theories I was completely unaware of.