Archive for neural network

adaptive copulas for ABC

Posted in Statistics with tags , , , , , , , , on March 20, 2019 by xi'an

A paper on ABC I read on my way back from Cambodia:  Yanzhi Chen and Michael Gutmann arXived an ABC [in Edinburgh] paper on learning the target via Gaussian copulas, to be presented at AISTATS this year (in Okinawa!). Linking post-processing (regression) ABC and sequential ABC. The drawback in the regression approach is that the correction often relies on an homogeneity assumption on the distribution of the noise or residual since this approach only applies a drift to the original simulated sample. Their method is based on two stages, a coarse-grained one where the posterior is approximated by ordinary linear regression ABC. And a fine-grained one, which uses the above coarse Gaussian version as a proposal and returns a Gaussian copula estimate of the posterior. This proposal is somewhat similar to the neural network approach of Papamakarios and Murray (2016). And to the Gaussian copula version of Li et al. (2017). The major difference being the presence of two stages. The new method is compared with other ABC proposals at a fixed simulation cost, which does not account for the construction costs, although they should be relatively negligible. To compare these ABC avatars, the authors use a symmetrised Kullback-Leibler divergence I had not met previously, requiring a massive numerical integration (although this is not an issue for the practical implementation of the method, which only calls for the construction of the neural network(s)). Note also that sequential ABC is only run for two iterations, and also that none of the importance sampling ABC versions of Fearnhead and Prangle (2012) and of Li and Fearnhead (2018) are considered, all versions relying on the same vector of summary statistics with a dimension much larger than the dimension of the parameter. Except in our MA(2) example, where regression does as well. I wonder at the impact of the dimension of the summary statistic on the performances of the neural network, i.e., whether or not it is able to manage the curse of dimensionality by ignoring all but essentially the data  statistics in the optimisation.

Bayesian intelligence in Warwick

Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on February 18, 2019 by xi'an

This is an announcement for an exciting CRiSM Day in Warwick on 20 March 2019: with speakers

10:00-11:00 Xiao-Li Meng (Harvard): “Artificial Bayesian Monte Carlo Integration: A Practical Resolution to the Bayesian (Normalizing Constant) Paradox”

11:00-12:00 Julien Stoehr (Dauphine): “Gibbs sampling and ABC”

14:00-15:00 Arthur Ulysse Jacot-Guillarmod (École Polytechnique Fedérale de Lausanne): “Neural Tangent Kernel: Convergence and Generalization of Deep Neural Networks”

15:00-16:00 Antonietta Mira (Università della Svizzera italiana e Università degli studi dell’Insubria): “Bayesian identifications of the data intrinsic dimensions”

[whose abstracts are on the workshop webpage] and free attendance. The title for the workshop mentions Bayesian Intelligence: this obviously includes human intelligence and not just AI!

a pen for ABC

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , on February 13, 2019 by xi'an

Among the flury of papers arXived around the ICML 2019 deadline, I read on my way back from Oxford a paper by Wiqvist et al. on learning summary statistics for ABC by neural nets. Pointing out at another recent paper by Jiang et al. (2017, Statistica Sinica) which constructed a neural network for predicting each component of the parameter vector based on the input (raw) data, as an automated non-parametric regression of sorts. Creel (2017) does the same but with summary statistics. The current paper builds up from Jiang et al. (2017), by adding the constraint that exchangeability and partial exchangeability features should be reflected by the neural net prediction function. With applications to Markovian models. Due to a factorisation theorem for d-block invariant models, the authors impose partial exchangeability for order d Markov models by combining two neural networks that end up satisfying this factorisation. The concept is exemplified for one-dimension g-and-k distributions, alpha-stable distributions, both of which are made of independent observations, and the AR(2) and MA(2) models, as in our 2012 ABC survey paper. Since the later is not Markovian the authors experiment with different orders and reach the conclusion that an order of 10 is most appropriate, although this may be impacted by being a ble to handle the true likelihood.

the $1,547.02 book on neural networks

Posted in Statistics with tags , , , , , on February 7, 2019 by xi'an

information maximising neural networks summaries

Posted in pictures, Statistics with tags , , , , , , , , on February 6, 2019 by xi'an

After missing the blood moon eclipse last night, I had a meeting today at the Paris observatory (IAP), where we discussed an ABC proposal made by Tom Charnock, Guilhem Lavaux, and Benjamin Wandelt from this institute.

“We introduce a simulation-based machine learning technique that trains artificial neural networks to find non-linear functionals of data that maximise Fisher information : information maximising neural networks.” T. Charnock et al., 2018
The paper is centred on the determination of “optimal” summary statistics. With the goal of finding “transformation which maps the data to compressed summaries whilst conserving Fisher information [of the original data]”. Which sounds like looking for an efficient summary and hence impossible in non-exponential cases. As seen from the description in (2.1), the assumed distribution of the summary is Normal, with mean μ(θ) and covariance matrix C(θ) that are implicit transforms of the parameter θ. In that respect, the approach looks similar to the synthetic likelihood proposal of Wood (2010). From which an unusual form of Fisher information can be derived, as μ(θ)’C(θ)⁻¹μ(θ)… A neural net is trained to optimise this information criterion at a given (so-called fiducial) value of θ, in terms of a set of summaries of the same dimension as the data. Which means the information contained in the whole data (likelihood) is not necessarily recovered, linking with this comment from Edward Ionides (in a set of lectures at Wharton).
“Even summary statistics derived by careful scientific or statistical reasoning have been found surprisingly uninformative compared to the whole data likelihood in both scientific investigations (Shrestha et al., 2011) and simulation experiments (Fasiolo et al., 2016)” E. Ionides, slides, 2017
The maximal Fisher information obtained in this manner is then used in a subsequent ABC step as the natural metric for the distance between the observed and simulated data. (Begging the question as to why being maximal is necessarily optimal.) Another question is about the choice of the fiducial parameter, which choice should be tested by for instance iterating the algorithm a few steps. But having to run simulations for a single value of the parameter is certainly a great selling point!

prepaid ABC

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , on January 16, 2019 by xi'an

Merijn Mestdagha, Stijn Verdoncka, Kristof Meersa, Tim Loossensa, and Francis Tuerlinckx from the KU Leuven, some of whom I met during a visit to its Wallon counterpart Louvain-La-Neuve, proposed and arXived a new likelihood-free approach based on saving simulations on a large scale for future users. Future users interested in the same model. The very same model. This makes the proposal quite puzzling as I have no idea as to when situations with exactly the same experimental conditions, up to the sample size, repeat over and over again. Or even just repeat once. (Some particular settings may accommodate for different sample sizes and the same prepaid database, but others as in genetics clearly do not.) I am sufficiently puzzled to suspect I have missed the message of the paper.

“In various fields, statistical models of interest are analytically intractable. As a result, statistical inference is greatly hampered by computational constraint s. However, given a model, different users with different data are likely to perform similar computations. Computations done by one user are potentially useful for other users with different data sets. We propose a pooling of resources across researchers to capitalize on this. More specifically, we preemptively chart out the entire space of possible model outcomes in a prepaid database. Using advanced interpolation techniques, any individual estimation problem can now be solved on the spot. The prepaid method can easily accommodate different priors as well as constraints on the parameters. We created prepaid databases for three challenging models and demonstrate how they can be distributed through an online parameter estimation service. Our method outperforms state-of-the-art estimation techniques in both speed (with a 23,000 to 100,000-fold speed up) and accuracy, and is able to handle previously quasi inestimable models.”

I foresee potential difficulties with this proposal, like compelling all future users to rely on the same summary statistics, on the same prior distributions (the “representative amount of parameter values”), and requiring a massive storage capacity. Plus furthermore relying at its early stage on the most rudimentary form of an ABC algorithm (although not acknowledged as such), namely the rejection one. When reading the description in the paper, the proposed method indeed selects the parameters (simulated from a prior or a grid) that are producing pseudo-observations that are closest to the actual observations (or their summaries s). The subsample thus constructed is used to derive a (local) non-parametric or machine-learning predictor s=f(θ). From which a point estimator is deduced by minimising in θ a deviance d(s⁰,f(θ)).

The paper does not expand much on the theoretical justifications of the approach (including the appendix that covers a formal situation where the prepaid grid conveniently covers the observed statistics). And thus does not explain on which basis confidence intervals should offer nominal coverage for the prepaid method. Instead, the paper runs comparisons with Simon Wood’s (2010) synthetic likelihood maximisation (Ricker model with three parameters), the rejection ABC algorithm (species dispersion trait model with four parameters), while the Leaky Competing Accumulator (with four parameters as well) seemingly enjoys no alternative. Which is strange since the first step of the prepaid algorithm is an ABC step, but I am unfamiliar with this model. Unsurprisingly, in all these cases, given that the simulation has been done prior to the computing time for the prepaid method and not for either synthetic likelihood or ABC, the former enjoys a massive advantage from the start.

“The prepaid method can be used for a very large number of observations, contrary to the synthetic likelihood or ABC methods. The use of very large simulated data sets allows investigation of large-sample properties of the estimator”

To return to the general proposal and my major reservation or misunderstanding, for different experiments, the (true or pseudo-true) value of the parameter will not be the same, I presume, and hence the region of interest [or grid] will differ. While, again, the computational gain is de facto obvious [since the costly production of the reference table is not repeated], and, to repeat myself, makes the comparison with methods that do require a massive number of simulations from scratch massively in favour of the prepaid option, I do not see a convenient way of recycling these prepaid simulations for another setting, that is, when some experimental factors, sample size or collection, or even just the priors, do differ. Again, I may be missing the point, especially in a specific context like repeated psychological experiments.

While this may have some applications in reproducibility (but maybe not, if the goal is in fact to detect cherry-picking), I see very little use in repeating the same statistical model on different datasets. Even repeating observations will require additional nuisance parameters and possibly perturb the likelihood and/or posterior to large extents.

Big Bayes goes South

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , on December 5, 2018 by xi'an

At the Big [Data] Bayes conference this week [which I found quite exciting despite a few last minute cancellations by speakers] there were a lot of clustering talks including the ones by Amy Herring (Duke), using a notion of centering that should soon appear on arXiv. By Peter Müller (UT, Austin) towards handling large datasets. Based on a predictive recursion that takes one value at a time, unsurprisingly similar to the update of Dirichlet process mixtures. (Inspired by a 1998 paper by Michael Newton and co-authors.) The recursion doubles in size at each observation, requiring culling of negligible components. Order matters? Links with Malsiner-Walli et al. (2017) mixtures of mixtures. Also talks by Antonio Lijoi and Igor Pruenster (Boconni Milano) on completely random measures that are used in creating clusters. And by Sylvia Frühwirth-Schnatter (WU Wien) on creating clusters for the Austrian labor market of the impact of company closure. And by Gregor Kastner (WU Wien) on multivariate factor stochastic models, with a video of a large covariance matrix evolving over time and catching economic crises. And by David Dunson (Duke) on distance clustering. Reflecting like myself on the definitely ill-defined nature of the [clustering] object. As the sample size increases, spurious clusters appear. (Which reminded me of a disagreement I had had with David McKay at an ICMS conference on mixtures twenty years ago.) Making me realise I missed the recent JASA paper by Miller and Dunson on that perspective.

Some further snapshots (with short comments visible by hovering on the picture) of a very high quality meeting [says one of the organisers!]. Following suggestions from several participants, it would be great to hold another meeting at CIRM in a near future. Continue reading