Abstract
In its full sense, perception rests on an agent's model of how its sensory input comes about and the inferences it draws based on this model. These inferences are necessarily uncertain. Here, we illustrate how the Hierarchical Gaussian Filter (HGF) offers a principled and generic way to deal with the several forms that uncertainty in perception takes. The HGF is a recent derivation of one-step update equations from Bayesian principles that rests on a hierarchical generative model of the environment and its (in)stability. It is computationally highly efficient, allows for online estimates of hidden states, and has found numerous applications to experimental data from human subjects. In this paper, we generalize previous descriptions of the HGF and its account of perceptual uncertainty. First, we explicitly formulate the extension of the HGF's hierarchy to any number of levels; second, we discuss how various forms of uncertainty are accommodated by the minimization of variational free energy as encoded in the update equations; third, we combine the HGF with decision models and demonstrate the inversion of this combination; finally, we report a simulation study that compared four optimization methods for inverting the HGF/decision model combination at different noise levels. These four methods (Nelder–Mead simplex algorithm, Gaussian process-based global optimization, variational Bayes and Markov chain Monte Carlo sampling) all performed well even under considerable noise, with variational Bayes offering the best combination of efficiency and informativeness of inference. Our results demonstrate that the HGF provides a principled, flexible, and efficient—but at the same time intuitive—framework for the resolution of perceptual uncertainty in behaving agents.
Highlights
Perception has long been proposed to take place in the context of prediction (Helmholtz, 1860)
For practical applications to empirical data, a general purpose modeling framework has been lacking that would allow for straightforward “off the shelf ” implementations of models explaining trial-wise empirical data from the Bayesian brain perspective. This is in contrast to reinforcement learning (RL) models which, due to their simplicity and computational efficiency, have found widespread application in experimental neuroscience, for example, in the analysis of functional magnetic resonance imaging and behavioral data. To fill this gap and provide a generic, robust and flexible framework for analysis of trial-wise data from the Bayesian brain perspective, we recently introduced the Hierarchical Gaussian Filter (HGF), a hierarchical Bayesian model Mathys et al (2011) in which states evolve as coupled Gaussian random walks, such that each state determines the step size of the evolution of the lower state
Our original formulation (Mathys et al, 2011) only contained three levels; here, we extend the HGF explicitly to any number of levels and show that the update equations maintain the same form across all levels because they are derived on the basis of the same coupling
Summary
Perception has long been proposed to take place in the context of prediction (Helmholtz, 1860) This entails that agents have a model of the environment which generates their sensory input. Probability theory formally prescribes how agents should learn about their environment from sensory information, given a model. This rests on sequential updating of beliefs according to Bayes’ theorem, where beliefs represent inferences about hidden states of the environment in the form of posterior probability distributions. It is this process that we refer to as perception. The second source of uncertainty is the possibility that states change with time, i.e., environmental uncertainty
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