On Kalman-Bucy filters, linear quadratic control and active inference

Baltieri, Manuel; Buckley, Christopher L.

doi:10.48550/arxiv.2005.06269

Cited by 5 publications

(8 citation statements)

References 41 publications

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“…Other works then argue that Kalman filters and variational free energy derivations in active inference are distinct and in some cases can be compared to each other [45,31,6,64,60,63,67,8,5,14,4], often claiming that active inference formulations outperform Kalman filters [45,31,6,64,60,67,14,4]. In [69,68] we then also find claims that approximations of Kalman filters derived from active inference are supposedly more biological plausible than their counterpart in standard filters.…”

Section: Related Workmentioning

confidence: 65%

“…In addition, our gradient-based approach preserves, more directly, (potential) connections that have been proposed between active inference and frameworks in, e.g., physics [34,9,59], chemistry [76], and brain science, specifically for neural dynamics [38,92,93,53,21]. Finally, this clarifies a series of claims about the relation between Kalman filters and active inference [45,31,44,6,64,58,11,33,29,35,28,43,36,37,2,3,39,71,15,1,60,80,30,42,81,40,23,41,63,78,52,34,79,77,67,19,10,75,62,8,…”

Section: Discussionmentioning

confidence: 82%

“…In this literature we find that only but a few works attempted to provide explicit formal connections via gradient descent on variational free energy [8,67,69,68], or via message passing on factor graphs [23,90]. Even fewer then have managed to correctly capture the actual equivalence [23,90], none of which using gradientbased algorithms.…”

Section: Related Workmentioning

confidence: 99%

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Kalman filters as the steady-state solution of gradient descent on variational free energy

Baltieri¹,

Isomura²

2021

Preprint

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The Kalman filter is an algorithm for the estimation of hidden variables in dynamical systems under linear Gauss-Markov assumptions with widespread applications across different fields. Recently, its Bayesian interpretation has received a growing amount of attention especially in neuroscience, robotics and machine learning. In neuroscience, in particular, models of perception and control under the banners of predictive coding, optimal feedback control, active inference and more generally the so-called Bayesian brain hypothesis, have all heavily relied on ideas behind the Kalman filter. Active inference, an algorithmic theory based on the free energy principle, specifically builds on approximate Bayesian inference methods proposing a variational account of neural computation and behaviour in terms of gradients of variational free energy. Using this ambitious framework, several works have discussed different possible relations between free energy minimisation and standard Kalman filters. With a few exceptions, however, such relations point at a mere qualitative resemblance or are built on a set of very diverse comparisons based on purported differences between free energy minimisation and Kalman filtering. In this work, we present a straightforward derivation of Kalman filters consistent with active inference via a variational treatment of free energy minimisation in terms of gradient descent. The approach considered here offers a more direct link between models of neural dynamics as gradient descent and standard accounts of perception and decision making based on probabilistic inference, further bridging the gap between hypotheses about neural implementation and computational principles in brain and behavioural sciences.

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Section: Related Workmentioning

confidence: 65%

Section: Discussionmentioning

confidence: 82%

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Kalman filters as the steady-state solution of gradient descent on variational free energy

Baltieri¹,

Isomura²

2021

Preprint

Self Cite

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“…Regarding more complex controllers, AIF shares similarities with Linear Quadratic Gaussian control (LQG), since both are grounded in Bayesian inference and optimal control [105]. However, a closer look reveals several key differences between the two approaches regarding the formulation of the state space, the cost functions, and their minimization-See [106].…”

Section: A Relationship With Classical Controllersmentioning

confidence: 99%

Active Inference in Robotics and Artificial Agents: Survey and Challenges

Lanillos¹,

Meo²,

Pezzato³

et al. 2021

Preprint

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Active inference is a mathematical framework which originated in computational neuroscience as a theory of how the brain implements action, perception and learning. Recently, it has been shown to be a promising approach to the problems of state-estimation and control under uncertainty, as well as a foundation for the construction of goal-driven behaviours in robotics and artificial agents in general. Here, we review the state-of-the-art theory and implementations of active inference for state-estimation, control, planning and learning; describing current achievements with a particular focus on robotics. We showcase relevant experiments that illustrate its potential in terms of adaptation, generalization and robustness. Furthermore, we connect this approach with other frameworks and discuss its expected benefits and challenges: a unified framework with functional biological plausibility using variational Bayesian inference.

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“…Our model is closely related to predictive coding approaches to brain function (Friston, 2005;Rao and Ballard, 1999;Spratling, 2017Spratling, , 2008. The predictive coding and Kalman Filtering update rules have been compared explicitly in the context of control theory (Baltieri and Buckley, 2020), although without any precise statement of their mathematical relationship. In some works (Friston et al, 2008;Friston, 2008), it has been claimed that predictive coding and Kalman filtering are equivalent.…”

Section: Related Workmentioning

confidence: 99%

Neural Kalman Filtering

Millidge,

Tschantz,

Seth

et al. 2021

Preprint

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The Kalman filter is a fundamental filtering algorithm that fuses noisy sensory data, a previous state estimate, and a dynamics model to produce a principled estimate of the current state. It assumes, and is optimal for, linear models and white Gaussian noise. Due to its relative simplicity and general effectiveness, the Kalman filter is widely used in engineering applications. Since many sensory problems the brain faces are, at their core, filtering problems, it is possible that the brain possesses neural circuitry that implements equivalent computations to the Kalman filter. The standard approach to Kalman filtering requires complex matrix computations that are unlikely to be directly implementable in neural circuits. In this paper, we show that a gradient-descent approximation to the Kalman filter requires only local computations with variance weighted prediction errors. Moreover, we show that it is possible under the same scheme to adaptively learn the dynamics model with a learning rule that corresponds directly to Hebbian plasticity. We demonstrate the performance of our method on a simple Kalman filtering task, and propose a neural implementation of the required equations.

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On Kalman-Bucy filters, linear quadratic control and active inference

Cited by 5 publications

References 41 publications

Kalman filters as the steady-state solution of gradient descent on variational free energy

Kalman filters as the steady-state solution of gradient descent on variational free energy

Active Inference in Robotics and Artificial Agents: Survey and Challenges

Neural Kalman Filtering

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