Hierarchical models in the brain.
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This means that a single model and optimisation scheme can be used to invert a wide range of models.We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data.We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.
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Affiliation: The Wellcome Trust Centre of Neuroimaging, University College London, London, United Kingdom. k.friston@fil.ion.ucl.ac.uk
ABSTRACT
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This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain. |
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Mentions: In this model, causes or inputs perturb the hidden states, which decayexponentially to produce an output that is a linear mixture of hiddenstates. Our example used a single input, two hidden states and four outputs.To generate data, we used a deterministic Gaussian bump function inputv(1) = exp(1/4(t−12)2)and the following parameters(50)During inversion, the cause is unknown and was subject tomildly informative (zero mean and unit precision) shrinkage priors. We alsotreated two of the parameters as unknown; one parameter from the observationfunction (the first) and one from the state equation (the second). Theseparameters had true values of 0.125 and −0.5, respectively, anduninformative shrinkage priors. The priors on the hyperparameters, sometimesreferred to as hyperpriors were similarly uninformative. These Gaussianhyperpriors effectively place lognormal hyperpriors on the precisions(strictly speaking, this invalidates the assumption of a linearhyperparameterisation but the effects are numerically small), because theprecisions scale as exp(λz) andexp(λw). Figure 5 shows a schematic of thegenerative model and the implicit recognition scheme based on predictionerrors. This scheme can be regarded as a message passing scheme that isconsidered in more depth in the next section. |
View Article: PubMed Central - PubMed
Affiliation: The Wellcome Trust Centre of Neuroimaging, University College London, London, United Kingdom. k.friston@fil.ion.ucl.ac.uk