Archive for unbiased estimation

a closed-form but intractable birthday

Posted in Books, Kids, pictures, Statistics with tags , , , , , on November 7, 2016 by xi'an

An interesting [at least for me!] question found on X Validated yesterday, namely how to simulate efficiently from the generalised birthday problem (or paradox) distribution, which provides the probability of finding exactly k different birthday dates as

\mathbb{P}(V = k) = \binom{n}{k}\displaystyle\sum_{i=0}^k (-1)^i \binom{k}{i}  \left(\frac{k-i}{n}\right)^m

where m is the number of individuals with a random birthday and n the number of days (e.g., n=365). The paradox with this closed-form formula (found by the inclusion-exclusion rule) is that it is too unstable to use per se. While it is always possible to run m draws from a uniform over {1,…,n} and count the number of different values, e.g.,

x=length(unique(sample(1:n,m,rep=TRUE)))

this takes much more time than using the exact distribution, if available:

sample(1:m,1e6,rep=TRUE,prob=eff[-1])

I played a little bit with the notion of using an unbiased estimator of the said probability, but the alternating series means that the unbiased estimator may end up being negative, which is an issue met in recent related papers like the famous Russian Roulette.

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…

MCqMC [#3]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , on August 20, 2016 by xi'an

On Thursday, Christoph Aistleiter [from TU Gräz] gave a plenary talk at MCqMC 2016 around Hermann Weyl’s 1916 paper, Über die Gleichverteilung von Zahlen mod. Eins, which demonstrates that the sequence a, 22a, 32a, … mod 1 is uniformly distributed on the unit interval when a is irrational. Obviously, the notion was not introduced for simulation purposes, but the construction applies in this setting! At least in a theoretical sense. Since for instance the result that the sequence (a,a²,a³,…) mod 1 being uniformly distributed for almost all a’s has not yet found one realisation a. But a nice hour of history of mathematics and number theory: it is not that common we hear the Riemann zeta function mentioned in a simulation conference!

The following session was a nightmare in that I wanted to attend all four at once! I eventually chose the transport session, in particular because Xiao-Li advertised it at the end of my talk. The connection is that his warp bridge sampling technique provides a folding map between modes of a target. Using a mixture representation of the target and folding all components to a single distribution. Interestingly, this transformation does not require a partition and preserves the normalising constants [which has a side appeal for bridge sampling of course]. In a problem with an unknown number of modes, the technique could be completed by [our] folding in order to bring the unobserved modes into the support of the folded target. Looking forward the incoming paper! The last talk of this session was by Matti Vihola, connecting multi-level Monte Carlo and unbiased estimation à la Rhee and Glynn, paper that I missed when it got arXived last December.

The last session of the day was about probabilistic numerics. I have already discussed extensively about this approach to numerical integration, to the point of being invited to the NIPS workshop as a skeptic! But this was an interesting session, both with introductory aspects and with new ones from my viewpoint, especially Chris Oates’ description of a PN method for handling both integrand and integrating measure as being uncertain. Another arXival that went under my decidedly deficient radar.

coupled filters

Posted in Kids, Statistics, University life with tags , , , , , , , , , on July 11, 2016 by xi'an

couplartPierre Jacob, Fredrik Lindsten, and Thomas Schön recently arXived a paper on coupled particle filters. A coupling problem that proves to be much more complicated than expected, due to the discrete nature of particle filters. The starting point of the paper is the use of common (e.g., uniform) random numbers for the generation of each entry in the particle system at each time t, which maximal correlation gets damaged by the resampling steps (even when using the same uniforms). One suggestion for improving the correlation between entries at each time made in the paper is to resort to optimal transport, using the distance between particles as the criterion. A cheaper alternative is inspired from multi-level Monte Carlo. It builds a joint multinomial distribution by optimising the coupling probability. [Is there any way to iterate this construct instead of considering only the extreme cases of identical values versus independent values?] The authors also recall a “sorted sampling” method proposed by Mike Pitt in 2002, which is to rely on the empirical cdfs derived from the particle systems and on the inverse cdf technique, which is the approach I would have first considered. Possibly with a smooth transform of both ecdf’s in order to optimise the inverse cdf move.  Actually, I have trouble with the notion that the ancestors of a pair of particles should matter. Unless one envisions a correlation of the entire path, but I am ensure how one can make paths correlated (besides coupling). And how this impacts likelihood estimation. As shown in the above excerpt, the coupled approximations produce regular versions and, despite the negative bias, fairly accurate evaluations of likelihood ratios, which is all that matters in an MCMC implementation. The paper also proposes a smoothing algorithm based on Rhee and Glynn (2012) debiasing technique, which operates on expectations against the smoothing distribution (conditional on a value of the parameter θ). Which may connect with the notion of simulating correlated paths. The interesting part is that, due to the coupling, the Rhee and Glynn unbiased estimator has a finite (if random) stopping time.

the penalty method

Posted in Statistics, University life with tags , , , , , , , , , , on July 7, 2016 by xi'an

“In this paper we will make conceptually simple generalization of Metropolis algorithm, by adjusting the acceptance ratio formula so that the transition probabilities are unaffected by the fluctuations in the estimate of [the acceptance ratio]…”

Last Friday, in Paris-Dauphine, my PhD student Changye Wu showed me a paper of Ceperley and Dewing entitled the penalty method for random walks with uncertain energies. Of which I was unaware of (and which alas pre-dated a recent advance made by Changye).  Despite its physics connections, the paper is actually about estimating a Metropolis-Hastings acceptance ratio and correcting the Metropolis-Hastings move for this estimation. While there is no generic solution to this problem, assuming that the logarithm of the acceptance ratio estimate is Gaussian around the true log acceptance ratio (and hence unbiased) leads to a log-normal correction for the acceptance probability.

“Unfortunately there is a serious complication: the variance needed in the noise penalty is also unknown.”

Even when the Gaussian assumption is acceptable, there is a further issue with this approach, namely that it also depends on an unknown variance term. And replacing it with an estimate induces further bias. So it may be that this method has not met with many followers because of those two penalising factors. Despite precluding the pseudo-marginal approach of Mark Beaumont (2003) by a few years, with the later estimating separately numerator and denominator in the Metropolis-Hastings acceptance ratio. And hence being applicable in a much wider collection of cases. Although I wonder if some generic approaches like path sampling or the exchange algorithm could be applied on a generic basis… [I just realised the title could be confusing in relation with the current football competition!]

auxiliary variable methods as ABC

Posted in Books, pictures, Statistics, University life with tags , , , , , on May 9, 2016 by xi'an

ruins of the abbey at Tynemouth, Sept. 03, 2013Dennis Prangle and Richard Everitt arXived a note today where they point out the identity between the auxiliary variable approach of Møller et al. (2006) [or rather its multiple or annealed version à la Murray] and [exact] ABC (as in our 2009 paper) in the case of Markov random fields. The connection between the two appears when using an importance sampling step in the ABC algorithm and running a Markov chain forward and backward the same number of steps as there are levels in the annealing scheme of MAV. Maybe more a curiosity than an indicator of a large phenomenon, since it is so rare that ABC can be use in its exact form.