Limits...
Uncertainty in perception and the Hierarchical Gaussian Filter.

Mathys CD, Lomakina EI, Daunizeau J, Iglesias S, Brodersen KH, Friston KJ, Stephan KE - Front Hum Neurosci (2014)

Bottom Line: It is computationally highly efficient, allows for online estimates of hidden states, and has found numerous applications to experimental data from human subjects.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.

View Article: PubMed Central - PubMed

Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London London, UK ; Max Planck UCL Centre for Computational Psychiatry and Ageing Research London, UK ; Translational Neuromodeling Unit, Institute for Biomedical Engineering, University of Zurich and ETH Zurich Zurich, Switzerland ; Laboratory for Social and Neural Systems Research (SNS Lab), Department of Economics, University of Zurich Zurich, Switzerland.

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.

No MeSH data available.


Related in: MedlinePlus

The 3-level HGF for binary outcomes. The lowest level, x1, is binary and corresponds, in the absence of sensory noise, to sensory input u. Left: schematic representation of the generative model as a Bayesian network. x(k)1, x(k)2, x(k)3 are hidden states of the environment at time point k. They generate u(k), the input at time point k, and depend on their immediately preceding values x(k − 1)2, x(k −1)3 and on the on parameters κ, ω, ϑ. Right: model definition. This figure has been adapted from Figures 1, 2 in Mathys et al. (2011).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4237059&req=5

Figure 2: The 3-level HGF for binary outcomes. The lowest level, x1, is binary and corresponds, in the absence of sensory noise, to sensory input u. Left: schematic representation of the generative model as a Bayesian network. x(k)1, x(k)2, x(k)3 are hidden states of the environment at time point k. They generate u(k), the input at time point k, and depend on their immediately preceding values x(k − 1)2, x(k −1)3 and on the on parameters κ, ω, ϑ. Right: model definition. This figure has been adapted from Figures 1, 2 in Mathys et al. (2011).

Mentions: In the three-level HGF for binary outcomes (Mathys et al., 2011) the third level, x3 is at the top, with constant step variance ϑ. The only level with a coupling of the form of Equation 5 is therefore the second level; this allows us to write κ2 ≡ κ and ω2 ≡ ω. We can allow for sensory uncertainty by including an additional level at the bottom of the hierarchy that predicts sensory input u from the state x1. In the absence of sensory uncertainty, knowledge of the state x1 enables accurate prediction of input u and vice versa; we may then simply set u ≡ x1 and treat x1 as if it were directly observed. A graphical overview of this model is given in Figure 2.


Uncertainty in perception and the Hierarchical Gaussian Filter.

Mathys CD, Lomakina EI, Daunizeau J, Iglesias S, Brodersen KH, Friston KJ, Stephan KE - Front Hum Neurosci (2014)

The 3-level HGF for binary outcomes. The lowest level, x1, is binary and corresponds, in the absence of sensory noise, to sensory input u. Left: schematic representation of the generative model as a Bayesian network. x(k)1, x(k)2, x(k)3 are hidden states of the environment at time point k. They generate u(k), the input at time point k, and depend on their immediately preceding values x(k − 1)2, x(k −1)3 and on the on parameters κ, ω, ϑ. Right: model definition. This figure has been adapted from Figures 1, 2 in Mathys et al. (2011).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4237059&req=5

Figure 2: The 3-level HGF for binary outcomes. The lowest level, x1, is binary and corresponds, in the absence of sensory noise, to sensory input u. Left: schematic representation of the generative model as a Bayesian network. x(k)1, x(k)2, x(k)3 are hidden states of the environment at time point k. They generate u(k), the input at time point k, and depend on their immediately preceding values x(k − 1)2, x(k −1)3 and on the on parameters κ, ω, ϑ. Right: model definition. This figure has been adapted from Figures 1, 2 in Mathys et al. (2011).
Mentions: In the three-level HGF for binary outcomes (Mathys et al., 2011) the third level, x3 is at the top, with constant step variance ϑ. The only level with a coupling of the form of Equation 5 is therefore the second level; this allows us to write κ2 ≡ κ and ω2 ≡ ω. We can allow for sensory uncertainty by including an additional level at the bottom of the hierarchy that predicts sensory input u from the state x1. In the absence of sensory uncertainty, knowledge of the state x1 enables accurate prediction of input u and vice versa; we may then simply set u ≡ x1 and treat x1 as if it were directly observed. A graphical overview of this model is given in Figure 2.

Bottom Line: It is computationally highly efficient, allows for online estimates of hidden states, and has found numerous applications to experimental data from human subjects.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.

View Article: PubMed Central - PubMed

Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London London, UK ; Max Planck UCL Centre for Computational Psychiatry and Ageing Research London, UK ; Translational Neuromodeling Unit, Institute for Biomedical Engineering, University of Zurich and ETH Zurich Zurich, Switzerland ; Laboratory for Social and Neural Systems Research (SNS Lab), Department of Economics, University of Zurich Zurich, Switzerland.

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.

No MeSH data available.


Related in: MedlinePlus