Archive for normalising constant

Another harmonic mean

Posted in Books, Statistics, University life with tags , , , , , , , , on May 21, 2022 by xi'an

Yet another paper that addresses the approximation of the marginal likelihood by a truncated harmonic mean, a popular theme of mine. A 2020 paper by Johannes Reich, entitled Estimating marginal likelihoods from the posterior draws through a geometric identity and published in Monte Carlo Methods and Applications.

The geometric identity it aims at exploiting is that

m(x) = \frac{\int_A \,\text d\theta}{\int_A \pi(\theta|x)\big/\pi(\theta)f(x|\theta)\,\text d\theta}

for any (positive volume) compact set $A$. This is exactly the same identity as in an earlier and uncited 2017 paper by Ana Pajor, with the also quite similar (!) title Estimating the Marginal Likelihood Using the Arithmetic Mean Identity and which I discussed on the ‘Og, linked with another 2012 paper by Lenk. Also discussed here. This geometric or arithmetic identity is again related to the harmonic mean correction based on a HPD region A that Darren Wraith and myself proposed at MaxEnt 2009. And that Jean-Michel and I presented at Frontiers of statistical decision making and Bayesian analysis in 2010.

In this avatar, the set A is chosen close to an HPD region, once more, with a structure that allows for an exact computation of its volume. Namely an ellipsoid that contains roughly 50% of the simulations from the posterior (rather than our non-intersecting union of balls centered at the 50% HPD points), which assumes a Euclidean structure of the parameter space (or, in other words, depends on the parameterisation)In the mixture illustration, the author surprisingly omits Chib’s solution, despite symmetrised versions avoiding the label (un)switching issues. . What I do not get is how this solution gets around the label switching challenge in that set A remains an ellipsoid for multimodal posteriors, which means it either corresponds to a single mode [but then how can a simulation be restricted to a “single permutation of the indicator labels“?] or it covers all modes but also the unlikely valleys in-between.


taking advantage of the constant

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , on May 19, 2022 by xi'an

A question from X validated had enough appeal for me to procrastinate about it for ½ an hour: what difference does it make [for simulation purposes] that a target density is properly normalised? In the continuous case, I do not see much to exploit about this knowledge, apart from the value potentially leading to a control variate (in a Gelfand and Dey 1996 spirit) and possibly to a stopping rule (by checking that the portion of the space visited so far has mass close to one, but this is more delicate than it sounds).

In a (possibly infinite) countable setting, it seems to me one gain (?) is that approximating expectations by Monte Carlo no longer requires iid simulations in the sense that once visited,  atoms need not be visited again. Self-avoiding random walks and their generalisations thus appear as a natural substitute for MC(MC) methods in this setting, provided finding unexplored atoms proves manageable. For instance, a stopping rule is always available, namely that the cumulated weight of the visited fraction of the space is close enough to one. The above picture shows a toy example on a 500 x 500 grid with 0.1% of the mass remaining at the almost invisible white dots. (In my experiment, neighbours for the random exploration were chosen at random over the grid, as I assumed no global information was available about the repartition over the grid either of mass function or of the function whose expectation was seeked.)

NCE, VAEs, GANs & even ABC…

Posted in Statistics with tags , , , , , , , , , , , , , on May 14, 2021 by xi'an

As I was preparing my (new) lectures for a PhD short course “at” Warwick (meaning on Teams!), I read a few surveys and other papers on all these acronyms. It included the massive Guttmann and Hyvärinen 2012 NCE JMLR paperGoodfellow’s NIPS 2016 tutorial on GANs, and  Kingma and Welling 2019 introduction to VAEs. Which I found a wee bit on the light side, maybe missing the fundamentals of the notion… As well as the pretty helpful 2019 survey on normalising flows by Papamakarios et al., although missing on the (statistical) density estimation side.  And also a nice (2017) survey of GANs by Shakir Mohamed and Balaji Lakshminarayanan with a somewhat statistical spirit, even though convergence issues are not again not covered. But misspecification is there. And the many connections between ABC and GANs, if definitely missing on the uncertainty aspects. While Deep Learning by Goodfellow, Bengio and Courville adresses both the normalising constant (or partition function) and GANs, it was somehow not deep enough (!) to use for the course, offering only a few pages on NCE, VAEs and GANs. (And also missing on the statistical references addressing the issue, incl. [or excl.]  Geyer, 1994.) Overall, the infinite variations offered on GANs leave me uncertain about their statistical relevance, as it is unclear how good the regularisation therein is for handling overfitting and consistent estimation. (And if I spot another decomposition of the Kullback-Leibler divergence, I may start crying…)

reXing the bridge

Posted in Books, pictures, Statistics with tags , , , , , , , , , on April 27, 2021 by xi'an

As I was re-reading Xiao-Li  Meng’s and Wing Hung Wong’s 1996 bridge sampling paper in Statistica Sinica, I realised they were making the link with Geyer’s (1994) mythical tech report, in the sense that the iterative construction of α functions “converges to the `reverse logistic regression’  described in Geyer (1994) for the two-density cases” (p.839). Although they also saw the later as an “iterative” application of Torrie and Valleau’s (1977) “umbrella sampling” estimator. And cited Bennett (1976) in the Journal of Computational Physics [for which Elsevier still asks for $39.95!] as the originator of the formula [check (6)]. And of the optimal solution (check (8)). Bennett (1976) also mentions that the method fares poorly when the targets do not overlap:

“When the two ensembles neither overlap nor satisfy the above smoothness condition, an accurate estimate of the free energy cannot be made without gathering additional MC data from one or more intermediate ensembles”

in which case this sequence of intermediate targets could be constructed and, who knows?!, optimised. (This may be the chain solution discussed in the conclusion of the paper.) Another optimisation not considered in enough detail is the allocation of the computing time to the two densities, maybe using a bandit strategy to avoid estimating the variance of the importance weights first.

flow contrastive estimation

Posted in Books, Statistics with tags , , , , , , , , on March 15, 2021 by xi'an

On the flight back from Montpellier, last week, I read a 2019 paper by Gao et al. revisiting the MLE estimation of a parametric family parameter when the normalising constant Z=Z(θ) is unknown. Via noise-contrastive estimation à la Guttman & Hyvärinnen (or à la Charlie Geyer). Treating the normalising constant Z as an extra parameter (as in Kong et al.) and the classification probability as an objective function and calling it a likelihood, which it is not in my opinion as (i) the allocation to the groups is not random and (ii) the original density of the actual observations does not appear in the so-called likelihood.

“When q appears on the right of KL-divergence [against p],  it is forced to cover most of the modes of p, When q appears on the left of KL-divergence, it tends to chase the major modes of p while ignoring the minor modes.”

The flow in the title indicates that the contrastive distribution q is estimated by a flow-based estimator, namely the transform of a basic noise distribution via easily invertible and differentiable transforms, for instance with lower triangular Jacobians. This flow is also estimated directly from the data but the authors complain this estimation is not good enough for noise contrastive estimation and suggest instead resorting to a GAN version where the classification log-probability is maximised in the model parameters and minimsed in the flow parameters. Except that I feel it misses the true likelihood part. In other words, why on Hyperion would estimating all θ, Z=Z(θ), and α at once improve the estimation of Z?

The other aspect that puzzles me is that (12) uses integrated classification probabilities (with the unknown Z as extra parameter), rather than conditioning on the data, Bayes-like. (The difference between (12) and GAN is that here the discriminator function is constrained.) Esp. when the first expectation is replaced with its empirical version.

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