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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This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies. This paper considers agency in the setting of embodied or active inference. In brief, we associate a sense of agency with prior beliefs about action and ask what sorts of beliefs underlie optimal behaviour. In particular, we consider prior beliefs that action minimises the Kullback-Leibler divergence between desired states and attainable states in the future. This allows one to formulate bounded rationality as approximate Bayesian inference that optimises a free energy bound on model evidence. We show that constructs like expected utility, exploration bonuses, softmax choice rules and optimism bias emerge as natural consequences of this formulation. Previous accounts of active inference have focused on predictive coding and Bayesian filtering schemes for minimising free energy. Here, we consider variational Bayes as an alternative scheme that provides formal constraints on the computational anatomy of inference and action – constraints that are remarkably consistent with neuroanatomy. Furthermore, this scheme contextualises optimal decision theory and economic (utilitarian) formulations as pure inference problems. For example, expected utility theory emerges as a special case of free energy minimisation, where the sensitivity or inverse temperature (of softmax functions and quantal response equilibria) has a unique and Bayes-optimal solution – that minimises free energy. This sensitivity corresponds to the precision of beliefs about behaviour, such that attainable goals are afforded a higher precision or confidence. In turn, this means that optimal behaviour entails a representation of confidence about outcomes that are under an agent's control. We consider the variational free energy approach for compressed sensing. We first show that the na "ive mean field approach performs remarkably well when coupled with a noise learning procedure. We also notice that it leads to the same equations as those used for iterative thresholding. We then discuss the Bethe free energy and how it corresponds to the fixed points of the approximate message passing algorithm. In both cases, we test numerically the direct optimization of the free energies as a converging sparse-estimationalgorithm. Prediction of three-dimensional structures of proteins and peptides by global optimization of the free energy estimate has been attempted without much success for over thirty years. The key problems were the insufficient accuracy of the free energy estimate and the giant size of the conformational space. Global optimization of the free energy function of a peptide in internal coordinate space is a powerful method of structure prediction that outperforms both Molecular Dynamics, bound by the continuity requirement, and Monte Carlo, bound by the Boltzmann ensemble generation requirement. We demonstrate that stochastic global optimization algorithms of the first order, i.e., with local minimization after each iteration (e.g., Monte Carlo-Minimization), have a greater chance of finding the global minimum after a fixed number of function evaluations. Recently, the principle of optimal bias was mathematically justified and the Optimal-Bias Monte Carlo-Minimization algorithm (a.k.a. Biased Probability Monte Carlo-minimization) was successfully applied to theoreticalab initiofolding of several peptides, resulting in more than a 10-fold increase in efficiency compared to the Monte Carlo-Minimization method. The square-root bias is shown to be comparable in performance with the previously derived linear bias. A 23-residue peptide of beta-beta-alpha structure can be predicted from any random starting conformation.