Archive for variance correction

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…