On the basis of computational studies it has been proposed that the central nervous system internally simulates the dynamic behavior of the motor system in planning, control, and learning; the existence and use of such an internal model is still under debate. A sensorimotor integration task was investigated in which participants estimated the location of one of their hands at the end of movements made in the dark and under externally imposed forces. The temporal propagation of errors in this task was analyzed within the theoretical framework of optimal state estimation. These results provide direct support for the existence of an internal model.

Abstract. This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models. We present a number of examples of graphical models, including the QMR-DT database, the sigmoid belief network, the Boltzmann machine, and several variants of hidden Markov models, in which it is infeasible to run exact inference algorithms. We then introduce variational methods, showing how upper and lower bounds can be found for local probabilities, and discussing methods for extending these bounds to bounds on global probabilities of interest. Finally we return to the examples and demonstrate how variational algorithms can be formulated in each case.

Unifying principles of movement have emerged from the computational study of motor control. We review several of these principles and show how they apply to processes such as motor planning, control, estimation, prediction and learning. Our goal is to demonstrate how specific models emerging from the computational approach provide a theoretical framework for movement neuroscience.

How can a machine learn from experience? Probabilistic modelling provides a framework for understanding what learning is, and has therefore emerged as one of the principal theoretical and practical approaches for designing machines that learn from data acquired through experience. The probabilistic framework, which describes how to represent and manipulate uncertainty about models and predictions, has a central role in scientific data analysis, machine learning, robotics, cognitive science and artificial intelligence. This Review provides an introduction to this framework, and discusses some of the state-of-the-art advances in the field, namely, probabilistic programming, Bayesian optimization, data compression and automatic model discovery.

In this paper we describe a scalable algorithm for the simultaneous mapping and localization (SLAM) problem. SLAM is the problem of acquiring a map of a static environment with a mobile robot. The vast majority of SLAM algorithms are based on the extended Kalman filter (EKF). In this paper we advocate an algorithm that relies on the dual of the EKF, the extended information filter (EIF). We show that when represented in the information form, map posteriors are dominated by a small number of links that tie together nearby features in the map. This insight is developed into a sparse variant of the EIF, called the sparse extended information filter (SEIF). SEIFs represent maps by graphical networks of features that are locally interconnected, where links represent relative information between pairs of nearby features, as well as information about the robot's pose relative to the map. We show that all essential update equations in SEIFs can be executed in constant time, irrespective of the size of the map. We also provide empirical results obtained for a benchmark data set collected in an outdoor environment, and using a multi-robot mapping simulation.

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Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observations and derivations made by many previous authors and introducing a new way of linking discrete and continuous state models using a simple nonlinearity. Through the use of other nonlinearities, we show how independent component analysis is also a variation of the same basic generative model. We show that factor analysis and mixtures of gaussians can be implemented in autoencoder neural networks and learned using squared error plus the same regularization term. We introduce a new model for static data, known as sensible principal component analysis, as well as a novel concept of spatially adaptive observation noise. We also review some of the literature involving global and local mixtures of the basic models and provide pseudocode for inference and learning for all the basic models. A Unifying ReviewMany common statistical techniques for modeling multidimensional static data sets and multidimensional time series can be seen as variants of one underlying model. As we will show, these include factor analysis, principal component analysis (PCA), mixtures of gaussian clusters, vector quantization, independent component analysis models (ICA), Kalman filter models (also known as linear dynamical systems), and hidden Markov models (HMMs). The relationships between some of these models has been noted in passing in the recent literature. For example, Hinton, Revow, and Dayan (1995) note that FA and PCA are closely related, and Digalakis, Rohlicek, and Ostendorf (1993) relate the forward-backward algorithm for HMMs to * Present address: {roweis, zoubin}@gatsby.ucl.ac.uk.

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