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.
翻译:卡尔曼过滤器是估算线性高斯-马尔科夫假设下动态系统中隐藏变量的算法,在不同领域广泛应用。最近,巴伊西亚解释得到越来越多的关注,特别是在神经科学、机器人和机器学习方面。特别是在神经科学方面,在预测编码、最佳反馈控制、积极推断以及更广义的所谓巴伊西亚大脑假设的旗帜下,感知和控制模式都严重依赖卡尔曼过滤器背后的想法。积极的推论,一种基于自由能源原则的算法理论,具体基于近似巴伊西亚的推论方法,从可变自由能源的梯度的梯度角度提出神经计算和行为的变异性账户。利用这一雄心勃勃勃的框架,若干著作讨论了自由能源最小化和标准卡尔曼过滤器之间的可能关系。然而,除了少数例外,这种关系点只是质相近,或者建立在基于所谓的自由能源最小性最小性稀释和Kalman过滤法之间差异的一套非常多样化的比较原则之上。在这项工作中,我们直截地展示了卡勒曼对神经计算和直位性递递的递递递的递性递度的递增性递增性模型,从而形成了一个基于最小性递化的递化的递化的递化的递化的递化的递增性模型。