Variational free energy and the Laplace approximation

By Karl J. Friston, Jérémie Mattout, Nelson J. Trujillo-Barreto, John Ashburner, W. D. Penny. 2007

947 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

A Comprehensive guide to Bayesian Convolutional Neural Network with Variational Inference (2019, positive): “Hence, we deﬁne the optimal parameters θ opt as where This derivation forms an optimisation problem with a resulting cost function widely known as variational free energy [50, 63, 12] which is built upon two terms: the former, KL [ q θ ( w”
Weight Uncertainty in Neural Networks (2015, background): “…θ q ( w | θ ) log q ( w | θ ) P ( w ) P ( D| w ) d w The resulting cost function is variously known as the variational free energy (Neal and Hinton, 1998; Yedidia et al., 2000; Friston et al., 2007) or the expected lower bound (Saul et al., 1996; Neal and Hinton, 1998; Jaakkola and Jordan, 2000).”
The functional anatomy of the MMN: A DCM study of the roving paradigm (2008, methodology): “Therefore, it allows for comparison of models with different numbers of parameters (Friston et al., 2006).”
A Free Energy Principle for Biological Systems (2012, methodology): “This is known as the Laplace assumption; see [14] for an introduction to predictive coding in visual neuroscience and [13] for an introduction to the Laplace assumption in the setting of variational free energy minimization.”
The free energy principle for action and perception: A mathematical review (2017, background): “Here nonlinear derivative terms such as µ′2, µ′µ′′, etc., have been neglected under a local linearity assumption (Friston et al., 2007) and only linear terms have been collected.”
A DCM for resting state fMRI (2014, methodology): “In our implementations we use Variational Laplace (Fristonetal.,2007)toevaluatemodelevidence p ( g ( ω )| m ) andthepos-terior density over model parameters p ( θ | g ( ω ), m ) in the usual way.”
Active Inference: A Process Theory (2017, background): “Technically, replacing the explicit solutions, equation 2.7, with a gradient ascent, equation 2.8, is exactly the same generalization of variational Bayes found in variational Laplace (Friston et al., 2007), namely, a generalized coordinate descent.”
A guide to group effective connectivity analysis, part 2: Second level analysis with PEB (2019, methodology): “The estimation scheme, called variational Laplace (Friston et al., 2007), tunes the parameters (e.”
Behavioral / Systems / Cognitive Striatal Prediction Error Modulates Cortical Coupling (2010, neutral): “For the nonlinear dynamic causal models of the fMRI data described below, we used the negative free-energy approximation to the log model evidence (Friston et al., 2007; Stephan et al., 2007b).”
Uncertainty in perception and the Hierarchical Gaussian Filter (2014, background): “Under the Laplace assumption (Friston et al., 2007), the negative variational free energy, an approximation to the log-model evidence, is − 196.19 for the ﬁrst response model and − 188.84 for the second.”
VBA: A Probabilistic Treatment of Nonlinear Models for Neurobiological and Behavioural Data (2014, methodology): “We refer to [9] for computational details about the VB algorithm.”
Bayesian Model Selection for Group Studies (2009, positive): “This is an example of a free-form VB approximation, where the optimal form of the approximate posterior (in this case a Dirichlet), has been derived rather than assumed before-hand (c.f. fixed-form VB approximations; Friston et al., 2007).”
Observing the Observer (II): Deciding When to Decide (2010, methodology): “Here, we use weakly informative log-normal priors (see [18]) on the perceptual parameter q and the response parameters h 1 , h 2 f g to enforce positivity.”
A free energy principle for a particular physics (2019, background): “s assumption leads to a particular formulation of Bayesian mechanics, where the (Fisher information) metric tensor is the precision matrix of the variational density – and the curvature of surprisal (Friston et al., 2007). Statistically speaking, a stipulative definition of the variational density –”
Bayesian Convolutional Neural Networks (2018, background): “This derivation forms an optimisation problem with a resulting cost function widely known as variational free energy [3, 17, 21] which is built upon two terms: the former is dependent on the definition of the prior p(w), thus called complexity cost, whereas the latter is dependent on the data p(D|w)”
A guide to group effective connectivity analysis, part 1: First level analysis with DCM for fMRI (2019, background): “Generally, model evidence cannot be calculated or derived analytically (because it involves marginalization over very high dimensional integrals); so instead an approximation called the negative variational free energy F (Friston et al., 2007) can be used.”