baby please don’t cry

First, an express riddle from the Riddler of last week:

An infant naps peacefully for two hours at a time and then wakes up, crying, due to hunger. After eating quickly, the infant plays alone for another hour, and then cries due to tiredness. This cycle repeats over the course of a 12-hour day. (The baby sleeps peacefully 12 hours through the night.) At a random time during the day, you spend 30 minutes with your baby and then the baby cries. What’s the probability that your baby is hungry?

The probabilistic setting is somewhat unclear, in particular because the last daytime nap is followed immediately with a 12 hour night sleep. Or the 12 hour night sleep is immediately followed by a one or two hour nap. Assuming a random starting time over the 12 hour period, denoting X as the time to the next crisis and Y as the nature of the cries (H versus T), it is straightforward to show that P(Y=H|X=30′) is ½. While it would be 1 for any duration larger than one hour.

Followed by an extra one this week:

Starting at a random time, 30 minutes go by with no cries. What is the probability that the next time your baby cries she will be hungry?

Which means computing P(Y=H|X>30′). Equal to ¾ in this case.

ABC in… everywhere [programme]

The ABC in Svalbard workshop is taking place on-line next week (and most sadly not in Svalbard). The programme is available on the ABC site. It starts (in Australia) at 4:00GMT (14 AEST) and finishes (in France) at 15:30GMT (17:30 CET). Registration is free but needed to access the Zoom codes!  See you on Zoom next week!!!

training energy based models

This recent arXival by Song and Kingma covers different computational approaches to semi-parametric estimation, but also exposes imho the chasm existing between statistical and machine learning perspectives on the problem.

“Energy-based models are much less restrictive in functional form: instead of specifying a normalized probability, they only specify the unnormalized negative log-probability (…) Since the energy function does not need to integrate to one, it can be parameterized with any nonlinear regression function.”

The above in the introduction appears first as a strange argument, since the mass one constraint is the least of the problems when addressing non-parametric density estimation. Problems like the convergence, the speed of convergence, the computational cost and the overall integrability of the estimator. It seems however that the restriction or lack thereof is to be understood as the ability to use much more elaborate forms of densities, which are then black-boxes whose components have little relevance… When using such mega-over-parameterised representations of densities, such as neural networks and normalising flows, a statistical assessment leads to highly challenging questions. But convergence (in the sample size) does not appear to be a concern for the paper. (Except for a citation of Hyvärinen on p.5.)

Using MLE in this context appears to be questionable, though, since the base parameter θ is not unlikely to remain identifiable. Computing the MLE is therefore a minor issue, in this regard, a resolution based on simulated gradients being well-chartered from the earlier era of stochastic optimisation as in Robbins & Monro (1954), Duflo (1996) or Benveniste & al. (1990). (The log-gradient of the normalising constant being estimated by the opposite of the gradient of the energy at a random point.)

“Running MCMC till convergence to obtain a sample x∼p(x) can be computationally expensive.”

Contrastive divergence à la Hinton (2002) is presented as a solution to the convergence problem by stopping early, which seems reasonable given the random gradient is mostly noise. With a possible correction for bias à la Jacob & al. (missing the published version).

An alternative to MLE is the 2005 Hyvärinen score, notorious for bypassing the normalising constant. But blamed in the paper for being costly in the dimension d of the variate x, due to the second derivative matrix. Which can be avoided by using Stein’s unbiased estimator of the risk (yay!) if using randomized data. And surprisingly linked with contrastive divergence as well, if a Taylor expansion is good enough an approximation! An interesting byproduct of the discussion on score matching is to turn it into an unintended form of ABC!

“Many methods have been proposed to automatically tune the noise distribution, such as Adversarial Contrastive Estimation (Bose et al., 2018), Conditional NCE (Ceylan and Gutmann, 2018) and Flow Contrastive Estimation (Gao et al., 2020).”

A third approach is the noise contrastive estimation method of Gutmann & Hyvärinen (2010) that connects with both others. And is a precursor of GAN methods, mentioned at the end of the paper via a (sort of) variational inequality.

ABC & the eighth plague of Egypt [locusts in forests]

“If you refuse to let them go, I will bring locusts into your country tomorrow. They will cover the face of the ground so that it cannot be seen. They will devour what little you have left after the hail, including every tree that is growing in your fields. They will fill your houses and those of all your officials and all the Egyptians.” Exodus 10:3-6

Marie-Pierre Chapuis, Louis Raynal, and co-authors, mostly from Montpellier, published last year a paper on the evolutionary history of the African arid-adapted pest locust, Schistocerca gregaria, called the eighth plague of Egypt in the Bible. And a cause for a major food disaster in East Africa over the past months. The analysis was run with ABC-RF techniques. The paper was first reviewed in PCI Evolutionary Biology, with the following points:

The present-day distribution of extant species is the result of the interplay between their past population demography (e.g., expansion, contraction, isolation, and migration) and adaptation to the environment (…) The understanding of the key factors driving species evolution gives important insights into how the species may respond to changing conditions, which can be particularly relevant for the management of harmful species, such as agricultural pests.

Meaningful demographic inferences present major challenges. These include formulating evolutionary scenarios fitting species biology and the eco-geographical context and choosing informative molecular markers and accurate quantitative approaches to statistically compare multiple demographic scenarios and estimate the parameters of interest. A further issue comes with result interpretation. Accurately dating the inferred events is far from straightforward since reliable calibration points are necessary to translate the molecular estimates of the evolutionary time into absolute time units (i.e. years). This can be attempted in different ways (…) Nonetheless, most experimental systems rarely meet these conditions, hindering the comprehensive interpretation of results.

The contribution of Chapuis et al. addresses these issues to investigate the recent history of the (…) desert locust (…) Owing to their fast mutation rate microsatellite markers offer at least two advantages: i) suitability for analyzing recently diverged populations, and ii) direct estimate of the germline mutation rate in pedigree samples (…) The main aim of the study is to infer the history of divergence of the two subspecies of the desert locust, which have spatially disjoint distribution corresponding to the dry regions of North and West-South Africa. They first use paleo-vegetation maps to formulate hypotheses about changes in species range since the last glacial maximum. Based on them, they generate 12 divergence models. For the selection of the demographic model and parameter estimation, they apply the recently developed ABC-RF approach (…) Some methodological novelties are also introduced in this work, such as the computation of the error associated with the posterior parameter estimates under the best scenario (…) The best-supported model suggests a recent divergence event of the subspecies of S. gregaria (around 2.6 kya) and a reduction of populations size in one of the subspecies (S. g. flaviventris) that colonized the southern distribution area. As such, results did not support the hypothesis that the southward colonization was driven by the expansion of African dry environments associated with the last glacial maximum (…) The estimated time of divergence points at a much more recent origin for the two subspecies, during the late Holocene, in a period corresponding to fairly stable arid conditions similar to current ones. Although the authors cannot exclude that their microsatellite data bear limited information on older colonization events than the last one, they bring arguments in favour of alternative explanations. The hypothesis privileged does not involve climatic drivers, but the particularly efficient dispersal behaviour of the species, whose individuals are able to fly over long distances (up to thousands of kilometers) under favourable windy conditions (…)

There is a growing number of studies in phylogeography in arid regions in the Southern hemisphere, but the impact of past climate changes on the species distribution in this region remains understudied relative to the Northern hemisphere. The study presented by Chapuis et al. offers several important insights into demographic changes and the evolutionary history of an agriculturally important pest species in Africa, which could also mirror the history of other organisms in the continent (…)

Microsatellite markers have been offering a useful tool in population genetics and phylogeography for decades (…) This study reaffirms the usefulness of these classic molecular markers to estimate past demographic events, especially when species- and locus-specific microsatellite mutation features are available and a powerful inferential approach is adopted. Nonetheless, there are still hurdles to overcome, such as the limitations in scenario choice associated with the simulation software used (e.g. not allowing for continuous gene flow in this particular case), which calls for further improvement of simulation tools allowing for more flexible modeling of demographic events and mutation patterns. In sum, this work not only contributes to our understanding of the makeup of the African biodiversity but also offers a useful statistical framework, which can be applied to a wide array of species and molecular markers.

 

 

Fisher, Bayes, and predictive Bayesian inference [seminar]

An interesting Foundations of Probability seminar at Rutgers University this Monday, at 4:30ET, 8:30GMT, by Sandy Zabell (the password is Angelina’s birthdate):

R. A. Fisher is usually perceived to have been a staunch critic of the Bayesian approach to statistics, yet his last book (Statistical Methods and Scientific Inference, 1956) is much closer in spirit to the Bayesian approach than the frequentist theories of Neyman and Pearson.  This mismatch between perception and reality is best understood as an evolution in Fisher’s views over the course of his life.  In my talk I will discuss Fisher’s initial and harsh criticism of “inverse probability”, his subsequent advocacy of fiducial inference starting in 1930, and his admiration for Bayes expressed in his 1956 book.  Several of the examples Fisher discusses there are best understood when viewed against the backdrop of earlier controversies and antagonisms.