**A** question on X validated gave me minor thought fodder for my crisp pre-dawn run in Warwick the other week: if one wants to use importance sampling for a variable Y that has no closed form density, but can be expressed as the transform (marginal) of a vector of variables with closed form densities, then, for Monte Carlo approximations, the problem can be reformulated as the computation of an integral of a transform of the vector itself and the importance ratio is given by the ratio of the true density of the vector over the density of the simulated vector. No Jacobian involved.

## Archive for importance sampling

## importance demarginalising

Posted in Books, Kids, pictures, Running, Statistics, Travel, University life with tags cross validated, importance sampling, Jacobian, Monte Carlo Statistical Methods, University of Warwick, Warwickshire on November 27, 2017 by xi'an## from least squares to signal processing and particle filtering

Posted in Books, Kids, Statistics, University life with tags Andrei Kolmogorov, auxiliary particle filter, Carl Friedrich Gauss, importance sampling, Kalman filter, R.E. Kálmán, Shannon, signal processing, stochastic processes on June 6, 2017 by xi'an**N**ozer Singpurwalla, Nick. Polson, and Refik Soyer have just arXived a remarkable survey on the history of signal processing, from Gauß, Yule, Kolmogorov and Wiener, to Ragazzini, Shanon, Kálmán [who, I was surprised to learn, died in Gainesville last year!], Gibbs sampling, and the particle filters of the 1990’s.

## nested sampling for philogenies

Posted in Statistics with tags estimating constants, evidence, harmonic mean estimator, importance sampling, nested sampling, normalising constant, philogenetic trees, philogenetics on March 3, 2017 by xi'an

“Methods to estimate the marginal likelihood should be sensitive to the prior choice. Non-informative priors should increase the contribution of low-likelihood regions of parameter space in the estimated marginal likelihood. Consequently, the prior choice should affect the estimated evidence.”

In a most recent arXival, Maturana, Brewer, and Klaere discuss of the appeal of nested sampling for conducting model choice in philogenetic models. In comparison with the “generalized steppingstone sampling” method, which represents the evidence as a product of ratios of evidences (Fan et al., 2011). And which I do not think I have previously met, with all references provided therein relating to Bayesian philogenetics, apparently. The stepping stone approach relies on a sequence of tempered targets, moving from a reference distribution to the real target as a temperature β goes from zero to one. (The paper also mentions thermodynamic integration as too costly.) Nested sampling—much discussed on this blog!—is presented in this paper as having the ability to deal with partly convex likelihoods, although I do not really get how or why. (As there is nothing new in the fairly pedagogical pretentation of nested sampling therein.) Nothing appears to be mentioned about the difficulty to handle multimodal as high likelihood isolated regions are unlikely to be sampled from poorly weighted priors (by which I mean that a region with significant likelihood mass is unlikely to get sampled if the prior distribution gives little prior weight to that region). The novelty in the paper is to compare nested sampling with generalized steppingstone sampling and path sampling on several phylogenetic examples. I did not spot computing time mentioned there. As usual with examples, my reservation is that the conclusions drawn for one particular analysis of one (three) particular example(s) does not convey a general method about the power and generality of a method. Even though I acknowledge that nested sampling is on principle operational in wide generality.

## scalable Langevin exact algorithm

Posted in Books, Statistics, Travel, University life with tags Brownian motion, control variate, importance sampling, JSM 2015, Langevin diffusion, normalising constant, Poisson process, quasi-stationary distribution, scalable MCMC, Seattle, sequential Monte Carlo, University of Warwick 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 Bernoulli, importance sampling, leave-one-out calibration, non-parametric kernel estimation, R, unbiased estimation, variance correction on October 13, 2016 by xi'an**F**ollowing 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 Bernoulli, importance sampling, leave-one-out calibration, non-parametric kernel estimation, unbiased estimation, variance correction on September 27, 2016 by xi'an**A**s 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

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:

- the integrand φ has a compact support, is bounded, and satisfies some Hölder-type regularity condition;
- the importance distribution ƒ is upper and lower bounded, its r-th order derivatives are upper bounded;
- the kernel K is order r, with exponential tails, and symmetric;
- the leave-one-out correction for bias has a cost O(n²) compared with O(n) cost of the regular Monte-Carlo estimator;
- 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 computing cost, Gaussian processes, importance sampling, log-normal distribution, Monte Carlo approximations, Monte Carlo Statistical Methods, scalability, sub-posteriors on June 8, 2016 by xi'an**C**hristopher 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:

- 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;
- 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?
- 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;
- The method relies on the exponentiation of a posterior expectation or simulation. As of yesterday, I am somehow wary of log-normal expectations!
- 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;
- The way the normalising missing constants and the duplicate simulations are processed (or not) could deserve further exploration;
- Computing costs are in fine unclear when compared with the other methods in the toolbox.