The predictive functions that permit humans to infer their body state by sensorimotor integration are critical to perform safe interaction in complex environments. These functions are adaptive and robust to non-linear actuators and noisy sensory information. This paper introduces a computational perceptual model based on predictive processing that enables any multisensory robot to learn, infer and update its body configuration when using arbitrary sensors with Gaussian additive noise. The proposed method integrates different sources of information (tactile, visual and proprioceptive) to drive the robot belief to its current body configuration. The motivation is to provide robots with the embodied perception needed for self-calibration and safe physical human-robot interaction. We formulate body learning as obtaining the forward model that encodes the sensor values depending on the body variables, and we solve it by Gaussian process regression. We model body estimation as minimizing the discrepancy between the robot body configuration belief and the observed posterior. We minimize the variational free energy using the sensory prediction errors (sensed vs expected). In order to evaluate the model we test it on a real multi-sensory robotic arm. We show how different sensor modalities contributions, included as additive errors, improve the refinement of the body estimation and how the system adapts itself to provide the most plausible solution even when injecting strong sensory visuo-tactile perturbations. We further analyse the reliability of the model when different sensor modalities are disabled. This provides grounded evidence about the correctness of the perceptual model and shows how the robot estimates and adjusts its body configuration just by means of sensory information.
The variational approach presented in this paper is related to expectation-maximization algorithms formalized as a maximization-maximization problem of the free-energy function @cite_16 . In fact, using Friston terminology, predictive processing is a dynamic expectation-maximization algorithm @cite_17 . Furthermore, it is important to highlight the strong similarities with the ensemble Kalman filter @cite_5 . We have adopted the free-energy mathematical framework for the following reasons: it provides indirect minimization of the Kullback-Leibler divergence @cite_17 ; it supports multisensory non-linear integration; it is scalable in its Laplace approximation @cite_30 ; it permits unsupervised parameters tuning @cite_20 ; and it is biologically plausible @cite_19 .
@cite_30 @cite_19 @cite_5 @cite_16 @cite_20 @cite_17
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The ‘free energy principle’ (FEP) has been suggested to provide a unified theory of the brain, integrating data and theory relating to action, perception, and learning. The theory and implementation of the FEP combines insights from Helmholtzian ‘perception as inference’, machine learning theory, and statistical thermodynamics. Here, we provide a detailed mathematical evaluation of a suggested biologically plausible implementation of the FEP that has been widely used to develop the theory. Our objectives are (i) to describe within a single article the mathematical structure of this implementation of the FEP; (ii) provide a simple but complete agent-based model utilising the FEP and (iii) to disclose the assumption structure of this implementation of the FEP to help elucidate its significance for the brain sciences. We describe a model of visual processing in which feedback connections from a higher- to a lower- order visual cortical area carry predictions of lower-level neural activities, whereas the feedforward connections carry the residual errors between the predictions and the actual lower-level activities. When exposed to natural images, a hierarchical network of model neurons implementing such a model developed simple-cell-like receptive fields. A subset of neurons responsible for carrying the residual errors showed endstopping and other extra-classical receptive-field effects. These results suggest that rather than being exclusively feedforward phenomena, nonclassical surround effects in the visual cortex may also result from cortico-cortical feedback as a consequence of the visual system using an efficient hierarchical strategy for encoding natural images. A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble. The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. This paper provides an easy to follow tutorial on the free-energy framework for modelling perception developed by Friston, which extends the predictive coding model of Rao and Ballard. These models assume that the sensory cortex infers the most likely values of attributes or features of sensory stimuli from the noisy inputs encoding the stimuli. Remarkably, these models describe how this inference could be implemented in a network of very simple computational elements, suggesting that this inference could be performed by biological networks of neurons. Furthermore, learning about the parameters describing the features and their uncertainty is implemented in these models by simple rules of synaptic plasticity based on Hebbian learning. This tutorial introduces the free-energy framework using very simple examples, and provides step-by-step derivations of the model. It also discusses in more detail how the model could be implemented in biological neural circuits. In particular, it presents an extended version of the model in which the neurons only sum their inputs, and synaptic plasticity only depends on activity of pre-synaptic and post-synaptic neurons. This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.