Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

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A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

Can machine learning deliver AI? Theoretical results, inspiration from the brain and cognition, as well as machine learning experiments suggest that in order to learn the kind of complicated functions that can represent high-level abstractions (e.g. in vision, language, and other AI-level tasks), one would need deep architectures. Deep architectures are composed of multiple levels of non-linear operations, such as in neural nets with many hidden layers, graphical models with many levels of latent variables, or in complicated propositional formulae re-using many sub-formulae. Each level of the architecture represents features at a different level of abstraction, defined as a composition of lower-level features. Searching the parameter space of deep architectures is a difficult task, but new algorithms have been discovered and a new sub-area has emerged in the machine learning community since 2006, following these discoveries. Learning algorithms such as those for Deep Belief Networks and other related unsupervised learning algorithms have recently been proposed to train deep architectures, yielding exciting results and beating the state-of-the-art in certain areas. Learning Deep Architectures for AI discusses the motivations for and principles of learning algorithms for deep architectures. By analyzing and comparing recent results with different learning algorithms for deep architectures, explanations for their success are proposed and discussed, highlighting challenges and suggesting avenues for future explorations in this area.

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Learning Deep Architectures for AI

https://www.sfs.uni-tuebingen.de/~hbaayen/publications/baayenDavidsonBates.pdf

Mixed-effects modeling with crossed random effects for subjects and items

We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem.

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Fast Computation of Wasserstein Barycenters

Approximate Bayesian computation (ABC) is a popular approach to address inference problems where the likelihood function is intractable, or expensive to calculate To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested Most effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples (Beaumont et al, Biometrika 86:983---990, 2009; Peters et al, Technical report, 2008; Toni et al, J Roy Soc Interface 6:187---202, 2009) and require the careful choice of simulation parameters In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and adaptively determines the simulation parameters We demonstrate our algorithm on a toy example and on a birth-death-mutation model arising in epidemiology

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An adaptive sequential Monte Carlo method for approximate Bayesian computation

In recent years particle filters have become a tremendouslypopular tool to perform tracking for non-linearand/or non-Gaussian models. This is due to their simplicity,generality and success over a wide range of challengingapplications. Particle filters, and Monte Carlo methodsin general, are however poor at consistently maintainingthe multi-modality of the target distributions that may arisedue to ambiguity or the presence of multiple objects. Toaddress this shortcoming this paper proposes to model thetarget distribution as a non-parametric mixture model, andpresents the general tracking recursion in this case. It isshown how a Monte Carlo implementation of the generalrecursion leads to a mixture of particle filters that interactonly in the computation of the mixture weights, thus leadingto an efficient numerical algorithm, where all the resultspertaining to standard particle filters apply. The ability ofthe new method to maintain posterior multi-modality is illustratedon a synthetic example and a real world trackingproblem involving the tracking of football players in a videosequence.

/pdf/maintaining-multi-modality-through-mixture-tracking-18a2in5buv.pdf

Maintaining Multi-Modality through Mixture Tracking

We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive models, are empirically more stable, generalize better and have a lower computational cost than Neural ODEs.

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Augmented Neural ODEs

Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.

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Arnaud Doucet

Papers

Fast Computation of Wasserstein Barycenters

An adaptive sequential Monte Carlo method for approximate Bayesian computation

Maintaining Multi-Modality through Mixture Tracking

Augmented Neural ODEs

On Particle Methods for Parameter Estimation in State-Space Models