Archive for unbiased estimation

multilevel Monte Carlo for estimating constants

Posted in Books, Statistics, University life with tags , , , , , on March 18, 2016 by xi'an

Pierre Del Moral, Ajay Jasra, Kody Law, and Yan Zhou just arXived a paper entitled Sequential Monte Carlo samplers for normalizing constants. Which obviously attracted my interest! The context is one of a sequential Monte Carlo problem, with an associated sequence of targets and of attached normalising constants. While the quantity of interest only relates to the final distribution in the sequence, using Mike Giles‘ multilevel Monte Carlo approach allows for a more accurate estimation and recycling all the past particles, thanks to the telescoping formula. And the sequential representation also allows for an unbiased estimator, as is well known in the sequential Monte Carlo literature. The paper derives accurate bounds on both the variances of two normalisation constant estimators and the costs of producing such estimators (assuming there is an index typo in Corollary 3.1, where L-2 should be L-1). The improvement when compared with traditional SMC is clear on the example contained in the paper. As I read the paper rather quickly and without much attention to the notations, I may have missed the point, but I did not see any conclusion on the choice of the particle population size at each iteration of the SMC. After asking Ajay about it, he pointed out that this size can be derived as

N_k=\epsilon^{-2}Lh_k^{(\beta+\zeta)/2}K_L

(with notations taken from the paper).

more e’s [and R’s]

Posted in Kids, pictures, R, Statistics with tags , , , , , , , on February 22, 2016 by xi'an

estimeAlex Thiéry suggested debiasing the biased estimate of e by Rhee and Glynn truncated series method, so I tried the method to see how much of an improvement (if any!) this would bring. I first attempted to naïvely implement the raw formula of Rhee and Glynn

\hat{\mathfrak{e}} = \sum_{n=1}^N \{\hat{e}_{n+1}-\hat{e}_n\}\big/\mathbb{P}(N\ge n)

with a (large) Poisson distribution on the stopping rule N, but this took ages. I then realised that the index n did not have to be absolute, i.e. to start at n=1 and proceed snailwise one integer at a time: the formula remains equally valid after a change of time, i.e. n=can start at an arbitrary value and proceeds by steps of arbitrary size, which obviously speeds things up! Continue reading

Bayesian model comparison with intractable constants

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on February 8, 2016 by xi'an

abcIRichard Everitt, Adam Johansen (Warwick), Ellen Rowing and Melina Evdemon-Hogan have updated [on arXiv] a survey paper on the computation of Bayes factors in the presence of intractable normalising constants. Apparently destined for Statistics and Computing when considering the style. A great entry, in particular for those attending the CRiSM workshop Estimating Constants in a few months!

A question that came to me from reading the introduction to the paper is why a method like Møller et al.’s (2006) auxiliary variable trick should be considered more “exact” than the pseudo-marginal approach of Andrieu and Roberts (2009) since the later can equally be seen as an auxiliary variable approach. The answer was on the next page (!) as it is indeed a special case of Andrieu and Roberts (2009). Murray et al. (2006) also belongs to this group with a product-type importance sampling estimator, based on a sequence of tempered intermediaries… As noted by the authors, there is a whole spectrum of related methods in this area, some of which qualify as exact-approximate, inexact approximate and noisy versions.

Their main argument is to support importance sampling as the method of choice, including sequential Monte Carlo (SMC) for large dimensional parameters. The auxiliary variable of Møller et al.’s (2006) is then part of the importance scheme. In the first toy example, a Poisson is opposed to a Geometric distribution, as in our ABC model choice papers, for which a multiple auxiliary variable approach dominates both ABC and Simon Wood’s synthetic likelihood for a given computing cost. I did not spot which artificial choice was made for the Z(θ)’s in both models, since the constants are entirely known in those densities. A very interesting section of the paper is when envisioning biased approximations to the intractable density. If only because the importance weights are most often biased due to the renormalisation (possibly by resampling). And because the variance derivations are then intractable as well. However, due to this intractability, the paper can only approach the impact of those approximations via empirical experiments. This leads however to the interrogation on how to evaluate the validity of the approximation in settings where truth and even its magnitude are unknown… Cross-validation and bootstrap type evaluations may prove too costly in realistic problems. Using biased solutions thus mostly remains an open problem in my opinion.

The SMC part in the paper is equally interesting if only because it focuses on the data thinning idea studied by Chopin (2002) and many other papers in the recent years. This made me wonder why an alternative relying on a sequence of approximations to the target with tractable normalising constants could not be considered. A whole sequence of auxiliary variable completions sounds highly demanding in terms of computing budget and also requires a corresponding sequence of calibrations. (Now, ABC fares no better since it requires heavy simulations and repeated calibrations, while further exhibiting a damning missing link with the target density. ) Unfortunately, embarking upon a theoretical exploration of the properties of approximate SMC is quite difficult, as shown by the strong assumptions made in the paper to bound the total variation distance to the true target.

love-hate Metropolis algorithm

Posted in Books, pictures, R, Statistics, Travel with tags , , , , , , , , , on January 28, 2016 by xi'an

Hyungsuk Tak, Xiao-Li Meng and David van Dyk just arXived a paper on a multiple choice proposal in Metropolis-Hastings algorithms towards dealing with multimodal targets. Called “A repulsive-attractive Metropolis algorithm for multimodality” [although I wonder why XXL did not jump at the opportunity to use the “love-hate” denomination!]. The proposal distribution includes a [forced] downward Metropolis-Hastings move that uses the inverse of the target density π as its own target, namely 1/{π(x)+ε}. Followed by a [forced] Metropolis-Hastings upward move which target is {π(x)+ε}. The +ε is just there to avoid handling ratios of zeroes (although I wonder why using the convention 0/0=1 would not work). And chosen as 10⁻³²³ by default in connection with R smallest positive number. Whether or not the “downward” move is truly downwards and the “upward” move is truly upwards obviously depends on the generating distribution: I find it rather surprising that the authors consider the same random walk density in both cases as I would have imagined relying on a more dispersed distribution for the downward move in order to reach more easily other modes. For instance, the downward move could have been based on an anti-Langevin proposal, relying on the gradient to proceed further down…

This special choice of a single proposal however simplifies the acceptance ratio (and keeps the overall proposal symmetric). The final acceptance ratio still requires a ratio of intractable normalising constants that the authors bypass by Møller et al. (2006) auxiliary variable trick. While the authors mention the alternative pseudo-marginal approach of Andrieu and Roberts (2009), they do not try to implement it, although this would be straightforward here: since the normalising constants are the probabilities of accepting a downward and an upward move, respectively. Those can easily be evaluated at a cost similar to the use of the auxiliary variables. That is,

– generate a few moves from the current value and record the proportion p of accepted downward moves;
– generate a few moves from the final proposed value and record the proportion q of accepted downward moves;

and replace the ratio of intractable normalising constants with p/q. It is not even clear that one needs those extra moves since the algorithm requires an acceptance in the downward and upward moves, hence generate Geometric variates associated with those probabilities p and q, variates that can be used for estimating them. From a theoretical perspective, I also wonder if forcing the downward and upward moves truly leads to an improved convergence speed. Considering the case when the random walk is poorly calibrated for either the downward or upward move, the number of failed attempts before an acceptance may get beyond the reasonable.

As XXL and David pointed out to me, the unusual aspect of the approach is that here the proposal density is intractable, rather than the target density itself. This makes using Andrieu and Roberts (2009) seemingly less straightforward. However, as I was reminded this afternoon at the statistics and probability seminar in Bristol, the argument for the pseudo-marginal based on an unbiased estimator is that w Q(w|x) has a marginal in x equal to π(x) when the expectation of w is π(x). In thecurrent problem, the proposal in x can extended into a proposal in (x,w), w P(w|x) whose marginal is the proposal on x.

If we complement the target π(x) with the conditional P(w|x), the acceptance probability would then involve

{π(x’) P(w’|x’) / π(x) P(w|x)} / {w’ P(w’|x’) / w P(w|x)} = {π(x’) / π(x)} {w/w’}

so it seems the pseudo-marginal (or auxiliary variable) argument also extends to the proposal. Here is a short experiment that shows no discrepancy between target and histogram:

nozero=1e-300
#love-hate move
move<-function(x){ 
  bacwa=1;prop1=prop2=rnorm(1,x,2) 
  while (runif(1)>{pi(x)+nozero}/{pi(prop1)+nozero}){ 
    prop1=rnorm(1,x,2);bacwa=bacwa+1}
  while (runif(1)>{pi(prop2)+nozero}/{pi(prop1)+nozero}) 
    prop2=rnorm(1,prop1,2)
  y=x
  if (runif(1)<pi(prop2)*bacwa/pi(x)/fowa){ 
    y=prop2;assign("fowa",bacwa)}
  return(y)}
#arbitrary bimodal target
pi<-function(x){.25*dnorm(x)+.75*dnorm(x,mean=5)}
#running the chain
T=1e5
x=5*rnorm(1);luv8=rep(x,T)
fowa=1;prop1=rnorm(1,x,2) #initial estimate
  while (runif(1)>{pi(x)+nozero}/{pi(prop1)+nozero}){
    fowa=fowa+1;prop1=rnorm(1,x,2)}
for (t in 2:T)
  luv8[t]=move(luv8[t-1])

exact ABC

Posted in Books, pictures, Statistics, University life with tags , , , , , on January 21, 2016 by xi'an

Sydney Opera from Sydney Harbour Bridge, Sydney, July 14, 2012Minh-Ngoc Tran and Robert Kohn have devised an “exact” ABC algorithm. They claim therein to remove the error due to the non-zero tolerance by using an unbiased estimator of the likelihood. Most interestingly, they start from the debiasing technique of Rhee and Glynn [also at the basis of the Russian roulette]. Which sums up as using a telescoping formula on a sequence of converging biased estimates. And cutting the infinite sum with a stopping rule.

“Our article proposes an ABC algorithm to estimate [the observed likelihood] that completely removes the error due to [the ABC] approximation…”

The sequence of biased but converging approximations is associated with a sequence of decreasing tolerances. The corresponding sequence of weights that determines the truncation in the series is connected to the decrease in the bias in an implicit manner for all realistic settings. Although Theorem 1 produces conditions on the ABC kernel and the sequence of tolerances and pseudo-sample sizes that guarantee unbiasedness and finite variance of the likelihood estimate. For a geometric stopping rule with rejection probability p, both tolerance and pseudo-sample size decrease as a power of p. As a side product the method also returns an unbiased estimate of the evidence. The overall difficulty I have with the approach is the dependence on the stopping rule and its calibration, and the resulting impact on the computing time of the likelihood estimate. When this estimate is used in a pseudo-marginal scheme à la Andrieu and Roberts (2009), I fear this requires new pseudo-samples at each iteration of the Metropolis-Hastings algorithm, which then becomes prohibitively expensive. Later today, Mark Girolami pointed out to me that Anne-Marie Lyne [one of the authors of the Russian roulette paper] also considered this exact approach in her thesis and concluded at an infinite computing time.

SPA 2015 Oxford [my day #2]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on July 17, 2015 by xi'an

KebleToday I [barely made it on a delayed train from Leaminton Spa to Oxford as I] chaired my invited session at SPA 2015 on advanced MCMC methodology. The three speakers, Randal Douc, Mike Pitt and Matti Vihola, all gave talks related to the pseudo-marginal technique. For instance, Randal gave examples of guaranteed variance improvements by adding randomisation steps in the generation of the rv’s behind the unbiased estimation of the likelihood function. Mike Pitt presented the paper I discussed a little while ago about evaluating the computing performances of pseudo-marginal approximations, with a fairly compelling perspective [I may have missed from the paper] on approximating the distribution on the approximation to the log-likelihood as a normal. Which led me to ponder at the ultimate version where the log-likelihood itself would get directly simulated in an MCMC algorithm bypassing the preliminary simulation of the parameters. Sounds a bit too fantasy-like to be of any use… Matti Vihola also presented recent results with Christophe Andrieu on comparing pseudo-marginal approximations, based on convex ordering properties. They included a domination result on ABC-MCM algorithms, as noted in a recent post. Which made me musing about the overall importance of unbiasedness in the global picture, where all we need are converging approximations, in fine.

Unbiased Bayes for Big Data: Path of partial posteriors [a reply from the authors]

Posted in Statistics, University life with tags , , , , , , , , , on February 27, 2015 by xi'an

[Here is a reply by Heiko Strathmann to my post of yesterday. Along with the slides of a talk in Oxford mentioned in the discussion.]

Thanks for putting this up, and thanks for the discussion. Christian, as already exchanged via email, here are some answers to the points you make.

First of all, we don’t claim a free lunch — and are honest with the limitations of the method (see negative examples). Rather, we make the point that we can achieve computational savings in certain situations — essentially exploiting redundancy (what Michael called “tall” data in his note on subsampling & HMC) leading to fast convergence of posterior statistics.

Dan is of course correct noticing that if the posterior statistic does not converge nicely (i.e. all data counts), then truncation time is “mammoth”. It is also correct that it might be questionable to aim for an unbiased Bayesian method in the presence of such redundancies. However, these are the two extreme perspectives on the topic. The message that we want to get along is that there is a trade-off in between these extremes. In particular the GP examples illustrate this nicely as we are able to reduce MSE in a regime where posterior statistics have *not* yet stabilised, see e.g. figure 6.

“And the following paragraph is further confusing me as it seems to imply that convergence is not that important thanks to the de-biasing equation.”

To clarify, the paragraph refers to the additional convergence issues induced by alternative Markov transition kernels of mini-batch-based full posterior sampling methods by Welling, Bardenet, Dougal & co. For example, Firefly MC’s mixing time is increased by a factor of 1/q where q*N is the mini-batch size. Mixing of stochastic gradient Langevin gets worse over time. This is not true for our scheme as we can use standard transition kernels. It is still essential for the partial posterior Markov chains to converge (if MCMC is used). However, as this is a well studied problem, we omit the topic in our paper and refer to standard tools for diagnosis. All this is independent of the debiasing device.

About MCMC convergence.
Yesterday in Oxford, Pierre Jacob pointed out that if MCMC is used for estimating partial posterior statistics, the overall result is not unbiased. We had a nice discussion how this bias could be addressed via a two-stage debiasing procedure: debiasing the MC estimates as described in the “Unbiased Monte Carlo” paper by Agapiou et al, and then plugging those into the path estimators — though it is (yet) not so clear how (and whether) this would work in our case.
In the current version of the paper, we do not address the bias present due to MCMC. We have a paragraph on this in section 3.2. Rather, we start from a premise that full posterior MCMC samples are a gold standard. Furthermore, the framework we study is not necessarily linked to MCMC – it could be that the posterior expectation is available in closed form, but simply costly in N. In this case, we can still unbiasedly estimate this posterior expectation – see GP regression.

“The choice of the tail rate is thus quite delicate to validate against the variance constraints (2) and (3).”

It is true that the choice is crucial in order to control the variance. However, provided that partial posterior expectations converge at a rate n with n the size of a minibatch, computational complexity can be reduced to N1-α (α<β) without variance exploding. There is a trade-off: the faster the posterior expectations converge, more computation can be saved; β is in general unknown, but can be roughly estimated with the “direct approach” as we describe in appendix.

About the “direct approach”
It is true that for certain classes of models and φ functionals, the direct averaging of expectations for increasing data sizes yields good results (see log-normal example), and we state this. However, the GP regression experiments show that the direct averaging gives a larger MSE as with debiasing applied. This is exactly the trade-off mentioned earlier.

I also wonder what people think about the comparison to stochastic variational inference (GP for Big Data), as this hasn’t appeared in discussions yet. It is the comparison to “non-unbiased” schemes that Christian and Dan asked for.

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