## vertical likelihood Monte Carlo integration

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

A few months ago, Nick Polson and James Scott arXived a paper on one of my favourite problems, namely the approximation of normalising constants (and it went way under my radar, as I only became aware of it quite recently!, then it remained in my travel bag for an extra few weeks…). The method for approximating the constant Z draws from an analogy with the energy level sampling methods found in physics, like the Wang-Landau algorithm. The authors rely on a one-dimensional slice sampling representation of the posterior distribution and [main innovation in the paper] add a weight function on the auxiliary uniform. The choice of the weight function links the approach with the dreaded harmonic estimator (!), but also with power-posterior and bridge sampling. The paper recommends a specific weighting function, based on a “score-function heuristic” I do not get. Further, the optimal weight depends on intractable cumulative functions as in nested sampling. It would be fantastic if one could draw directly from the prior distribution of the likelihood function—rather than draw an x [from the prior or from something better, as suggested in our 2009 Biometrika paper] and transform it into L(x)—but as in all existing alternatives this alas is not the case. (Which is why I find the recommendations in the paper for practical implementation rather impractical, since, were the prior cdf of L(X) available, direct simulation of L(X) would be feasible. Maybe not the optimal choice though.)

“What is the distribution of the likelihood ordinates calculated via nested sampling? The answer is surprising: it is essentially the same as the distribution of likelihood ordinates by recommended weight function from Section 4.”

The approach is thus very much related to nested sampling, at least in spirit. As the authors later demonstrate, nested sampling is another case of weighting, Both versions require simulations under truncated likelihood values. Albeit with a possibility of going down [in likelihood values] with the current version. Actually, more weighting could prove [more] efficient as both the original nested and vertical sampling simulate from the prior under the likelihood constraint. Getting away from the prior should help. (I am quite curious to see how the method is received and applied.)

## reis naar Amsterdam

Posted in Books, Kids, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , on April 16, 2015 by xi'an

On Monday, I went to Amsterdam to give a seminar at the University of Amsterdam, in the department of psychology. And to visit Eric-Jan Wagenmakers and his group there. And I had a fantastic time! I talked about our mixture proposal for Bayesian testing and model choice without getting hostile or adverse reactions from the audience, quite the opposite as we later discussed this new notion for several hours in the café across the street. I also had the opportunity to meet with Peter Grünwald [who authored a book on the minimum description length principle] pointed out a minor inconsistency of the common parameter approach, namely that the Jeffreys prior on the first model did not have to coincide with the Jeffreys prior on the second model. (The Jeffreys prior for the mixture being unavailable.) He also wondered about a more conservative property of the approach, compared with the Bayes factor, in the sense that the non-null parameter could get closer to the null-parameter while still being identifiable.

Among the many persons I met in the department, Maarten Marsman talked to me about his thesis research, Plausible values in statistical inference, which involved handling the Ising model [a non-sparse Ising model with O(p²) parameters] by an auxiliary representation due to Marc Kac and getting rid of the normalising (partition) constant by the way. (Warning, some approximations involved!) And who showed me a simple probit example of the Gibbs sampler getting stuck as the sample size n grows. Simply because the uniform conditional distribution on the parameter concentrates faster (in 1/n) than the posterior (in 1/√n). This does not come as a complete surprise as data augmentation operates in an n-dimensional space. Hence it requires more time to get around. As a side remark [still worth printing!], Maarten dedicated his thesis as “To my favourite random variables , Siem en Fem, and to my normalizing constant, Esther”, from which I hope you can spot the influence of at least two of my book dedications! As I left Amsterdam on Tuesday, I had time for a enjoyable dinner with E-J’s group, an equally enjoyable early morning run [with perfect skies for sunrise pictures!], and more discussions in the department. Including a presentation of the new (delicious?!) Bayesian software developed there, JASP, which aims at non-specialists [i.e., researchers unable to code in R, BUGS, or, God forbid!, STAN] And about the consequences of mixture testing in some psychological experiments. Once again, a fantastic time discussing Bayesian statistics and their applications, with a group of dedicated and enthusiastic Bayesians!

## density normalization for MCMC algorithms

Posted in Statistics, University life with tags , , , , , , , , on November 6, 2014 by xi'an

Another paper addressing the estimation of the normalising constant and the wealth of available solutions just came out on arXiv, with the full title of “Target density normalization for Markov chain Monte Carlo algorithms“, written by Allen Caldwell and Chang Liu. (I became aware of it by courtesy of Ewan Cameron, as it appeared in the physics section of arXiv. It is actually a wee bit annoying that papers in the subcategory “Data Analysis, Statistics and Probability” of physics do not get an automated reposting on the statistics lists…)

In this paper, the authors compare three approaches to the problem of finding

$\mathfrak{I} = \int_\Omega f(\lambda)\,\text{d}\lambda$

when the density f is unormalised, i.e., in more formal terms, when f is proportional to a probability density (and available):

1. an “arithmetic mean”, which is an importance sampler based on (a) reducing the integration volume to a neighbourhood ω of the global mode. This neighbourhood is chosen as an hypercube and the importance function turns out to be the uniform over this hypercube. The corresponding estimator is then a rescaled version of the average of f over uniform simulations in ω.
2.  an “harmonic mean”, of all choices!, with again an integration over the neighbourhood ω of the global mode in order to avoid the almost sure infinite variance of harmonic mean estimators.
3. a Laplace approximation, using the target at the mode and the Hessian at the mode as well.

The paper then goes to comparing those three solutions on a few examples, demonstrating how the diameter of the hypercube can be calibrated towards a minimum (estimated) uncertainty. The rather anticlimactic conclusion is that the arithmetic mean is the most reliable solution as harmonic means may fail in larger dimension and more importantly fail to signal its failure, while Laplace approximations only approximate well quasi-Gaussian densities…

What I find most interesting in this paper is the idea of using only one part of the integration space to compute the integral, even though it is not exactly new. Focussing on a specific region ω has pros and cons, the pros being that the reduction to a modal region reduces needs for absolute MCMC convergence and helps in selecting alternative proposals and also prevents from the worst consequences of using a dreaded harmonic mean, the cons being that the region needs be well-identified, which means requirements on the MCMC kernel, and that the estimate is a product of two estimates, the frequency being driven by a Binomial noise.  I also like very much the idea of calibrating the diameter Δof the hypercube ex-post by estimating the uncertainty.

As an aside, the paper mentions most of the alternative solutions I just presented in my Monte Carlo graduate course two days ago (like nested or bridge or Rao-Blackwellised sampling, including our proposal with Darren Wraith), but dismisses them as not “directly applicable in an MCMC setting”, i.e., without modifying this setting. I unsurprisingly dispute this labelling, both because something like the Laplace approximation requires extra-work on the MCMC output (and once done this work can lead to advanced Laplace methods like INLA) and because other methods could be considered as well (for instance, bridge sampling over several hypercubes). As shown in the recent paper by Mathieu Gerber and Nicolas Chopin (soon to be discussed at the RSS!), MCqMC has also become a feasible alternative that would compete well with the methods studied in this paper.

Overall, this is a paper that comes in a long list of papers on constant approximations. I do not find the Markov chain of MCMC aspect particularly compelling or specific, once the effective sample size is accounted for. It would be nice to find generic ways of optimising the visit to the hypercube ω and to estimate efficiently the weight of ω. The comparison is solely run over examples, but they all rely on a proper characterisation of the hypercube and the ability to simulate efficiently f over that hypercube.

## independent component analysis and p-values

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , on September 8, 2014 by xi'an

Last morning at the neuroscience workshop Jean-François Cardoso presented independent component analysis though a highly pedagogical and enjoyable tutorial that stressed the geometric meaning of the approach, summarised by the notion that the (ICA) decomposition

$X=AS$

of the data X seeks both independence between the columns of S and non-Gaussianity. That is, getting as away from Gaussianity as possible. The geometric bits came from looking at the Kullback-Leibler decomposition of the log likelihood

$-\mathbb{E}[\log L(\theta|X)] = KL(P,Q_\theta) + \mathfrak{E}(P)$

where the expectation is computed under the true distribution P of the data X. And Qθ is the hypothesised distribution. A fine property of this decomposition is a statistical version of Pythagoreas’ theorem, namely that when the family of Qθ‘s is an exponential family, the Kullback-Leibler distance decomposes into

$KL(P,Q_\theta) = KL(P,Q_{\theta^0}) + KL(Q_{\theta^0},Q_\theta)$

where θ⁰ is the expected maximum likelihood estimator of θ. (We also noticed this possibility of a decomposition in our Kullback-projection variable-selection paper with Jérôme Dupuis.) The talk by Aapo Hyvärinen this morning was related to Jean-François’ in that it used ICA all the way to a three-level representation if oriented towards natural vision modelling in connection with his book and the paper on unormalised models recently discussed on the ‘Og.

On the afternoon, Eric-Jan Wagenmaker [who persistently and rationally fight the (ab)use of p-values and who frequently figures on Andrew’s blog] gave a warning tutorial talk about the dangers of trusting p-values and going fishing for significance in existing studies, much in the spirit of Andrew’s blog (except for the defence of Bayes factors). Arguing in favour of preregistration. The talk was full of illustrations from psychology. And included the line that ESP testing is the jester of academia, meaning that testing for whatever form of ESP should be encouraged as a way to check testing procedures. If a procedure finds a significant departure from the null in this setting, there is something wrong with it! I was then reminded that Eric-Jan was one of the authors having analysed Bem’s controversial (!) paper on the “anomalous processes of information or energy transfer that are currently unexplained in terms of known physical or biological mechanisms”… (And of the shocking talk by Jessica Utts on the same topic I attended in Australia two years ago.)

## this issue of Series B

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , on September 5, 2014 by xi'an

The September issue of [JRSS] Series B I received a few days ago is of particular interest to me. (And not as an ex-co-editor since I was never involved in any of those papers!) To wit: a paper by Hani Doss and Aixin Tan on evaluating normalising constants based on MCMC output, a preliminary version I had seen at a previous JSM meeting, a paper by Nick Polson, James Scott and Jesse Windle on the Bayesian bridge, connected with Nick’s talk in Boston earlier this month, yet another paper by Ariel Kleiner, Ameet Talwalkar, Purnamrita Sarkar and Michael Jordan on the bag of little bootstraps, which presentation I heard Michael deliver a few times when he was in Paris. (Obviously, this does not imply any negative judgement on the other papers of this issue!)

For instance, Doss and Tan consider the multiple mixture estimator [my wording, the authors do not give the method a name, referring to Vardi (1985) but missing the connection with Owen and Zhou (2000)] of k ratios of normalising constants, namely

$\sum_{l=1}^k \frac{1}{n_l} \sum_{t=1}^{n_l} \dfrac{n_l g_j(x_t^l)}{\sum_{s=1}^k n_s g_s(x_t^l) z_1/z_s } \longrightarrow \dfrac{z_j}{z_1}$

where the z’s are the normalising constants and with possible different numbers of iterations of each Markov chain. An interesting starting point (that Hans Künsch had mentioned to me a while ago but that I had since then forgotten) is that the problem was reformulated by Charlie Geyer (1994) as a quasi-likelihood estimation where the ratios of all z’s relative to one reference density are the unknowns. This is doubling interesting, actually, because it restates the constant estimation problem into a statistical light and thus somewhat relates to the infamous “paradox” raised by Larry Wasserman a while ago. The novelty in the paper is (a) to derive an optimal estimator of the ratios of normalising constants in the Markov case, essentially accounting for possibly different lengths of the Markov chains, and (b) to estimate the variance matrix of the ratio estimate by regeneration arguments. A favourite tool of mine, at least theoretically as practically useful minorising conditions are hard to come by, if at all available.

## where did the normalising constants go?! [part 2]

Posted in R, Statistics, Travel with tags , , , , , , , on March 12, 2014 by xi'an

Coming (swiftly and smoothly) back home after this wonderful and intense week in Banff, I hugged my loved ones,  quickly unpacked, ran a washing machine, and  then sat down to check where and how my reasoning was wrong. To start with, I experimented with a toy example in R:

# true target is (x^.7(1-x)^.3) (x^1.3 (1-x)^1.7)
# ie a Beta(3,3) distribution

# samples from partial posteriors
N=10^5
sam1=rbeta(N,1.7,1.3)
sam2=rbeta(N,2.3,2.7)

# first version: product of density estimates
dens1=density(sam1,from=0,to=1)
dens2=density(sam2,from=0,to=1)
prod=dens1$y*dens2$y
# normalising by hand
prod=prod*length(dens1$x)/sum(prod) plot(dens1$x,prod,type="l",col="steelblue",lwd=2)

# second version: F-S & P's yin+yang sampling
# with weights proportional to the other posterior

subsam1=sam1[sample(1:N,N,prob=dbeta(sam1,2.3,2.7),rep=T)]
plot(density(subsam1,from=0,to=1),col="steelblue",lwd=2)

subsam2=sam2[sample(1:N,N,prob=dbeta(sam2,1.7,1.3),rep=T)]
plot(density(subsam2,from=0,to=1),col="steelblue",lwd=2)


and (of course!) it produced the perfect fits reproduced below. Writing the R code acted as a developing bath as it showed why we could do without the constants!

Of course”, because the various derivations in the above R code all are clearly independent from the normalising constant: (i) when considering a product of kernel density estimators, as in the first version, this is an approximation of

$\prod_{i=1}^k p_i(\theta)$

as well as of

$\prod_{ i}^k m_i(\theta)$

since the constant does not matter. (ii) When considering a sample from mi and weighting it by the product of the remaining true or estimated mj‘s, this is a sampling weighting resampling simulation from the density proportional to the product and hence, once again, the constants do not matter. At last, (iii) when mixing the two subsamples, since they both are distributed from the product density, the constants do not matter. As I slowly realised when running this morning (trail-running, not code-runninh!, for the very first time in ten days!), the straight-from-the-box importance sampling version on the mixed samples I considered yesterday (to the point of wondering out loud where did the constants go) is never implemented in the cited papers. Hence, the fact that

$\prod_i^k p_i(\theta)\propto \prod_{i}^k m_i(\theta)$

is enough to justify handling the target directly as the product of the partial marginals. End of the mystery. Anticlimactic end, sorry…

## where did the normalising constants go?! [part 1]

Posted in R, Statistics, Travel with tags , , , , on March 11, 2014 by xi'an

When listening this week to several talks in Banff handling large datasets or complex likelihoods by parallelisation, splitting the posterior as

$\prod_{i=1}^k p_i(\theta)$

and handling each term of this product on a separate processor or thread as proportional to a probability density,

$p_i(\theta)\propto m_i(\theta)=\omega_i p_i(\theta),$

then producing simulations from the mi‘s and attempting at deriving simulations from the original product, I started to wonder where all those normalising constants went. What vaguely bothered me for a while, even prior to the meeting, and then unclicked thanks to Sylvia’s talk yesterday was the handling of the normalising constants ωi by those different approaches… Indeed, it seemed to me that the samples from the mi‘s should be weighted by

$\omega_i\prod_{j\ne i}^k p_j(\theta)$

rather than just

$\prod_{j\ne i}^k p_j(\theta)$

or than the product of the other posteriors

$\prod_{j\ne i}^k m_j(\theta)$

which makes or should make a significant difference. For instance, a sheer importance sampling argument for the aggregated sample exhibited those weights

$\mathbb{E}[h(\theta_i)\prod_{i=1}^k p_i(\theta_i)\big/m_i(\theta_i)]=\omega_i^{-1}\int h(\theta_i)\prod_{j}^k p_j(\theta_i)\text{d}\theta_i$

Hence processing the samples on an equal footing or as if the proper weight was the product of the other posteriors mj should have produced a bias in the resulting sample. This was however the approach in both Scott et al.‘s and Neiswanger et al.‘s perspectives. As well as Wang and Dunson‘s, who also started from the product of posteriors. (Normalizing constants are considered in, e.g., Theorem 1, but only for the product density and its Weierstrass convolution version.) And in Sylvia’s talk. Such a consensus of high calibre researchers cannot get it wrong! So I must have missed something: what happened is that the constants eventually did not matter, as expanded in the next post