## Variational filtering (2008)

Citations: | 7 - 4 self |

### BibTeX

@MISC{Friston08variationalfiltering,

author = {K. J. Friston},

title = {Variational filtering},

year = {2008}

}

### OpenURL

### Abstract

This note presents a simple Bayesian filtering scheme, using variational calculus, for inference on the hidden states of dynamic systems. Variational filtering is a stochastic scheme that propagates particles over a changing variational energy landscape, such that their sample density approximates the conditional density of hidden and states and inputs. The key innovation, on which variational filtering rests, is a formulation in generalised coordinates of motion. This renders the scheme much simpler and more versatile than existing approaches, such as those based on particle filtering. We demonstrate variational filtering using simulated and real data from hemodynamic systems studied in neuroimaging and provide comparative evaluations using particle filtering and the fixed-form homologue of variational filtering, namely dynamic expectation maximisation.

### Citations

1147 | A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking
- Arulampalam, Maskell, et al.
(Show Context)
Citation Context ...relation to filtering. A nonlinear convolution model Here, we focus on the effect of nonlinearities with a model that has been used previously to compare extended Kalman and particle filtering (c.f., =-=Arulampalam et al., 2002-=-) Nonlinear (double-well) convolution model Fig. 6. Schematic detailing the nonlinear convolution model in which hidden states evolve in a double-well potential. (a): Plot of the velocity of states ag... |

330 | The Theory of Stochastic Processes - Cox, Miller - 1965 |

180 |
Dynamic causal modelling
- Friston, Harrison, et al.
- 2003
(Show Context)
Citation Context ...(fMRI). This example has been chosen because inference about brain states from non-invasive neurophysiologic observations is an important issue in cognitive neuroscience and functional imaging (e.g., =-=Friston et al., 2003-=-; Gitelman et al., 2003; Buxton et al., 2004; Riera et al., 2004; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communi... |

133 |
Dynamics of blood flow and oxygenation changes during brain activation: the Balloon model
- Buxton, Wong, et al.
- 1998
(Show Context)
Citation Context ...n et al., 2003; Buxton et al., 2004; Riera et al., 2004; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communications (=-=Buxton et al., 1998-=-; Friston, 2002). In brief, neuronal activity causes an increase in a vasodilatory signal h 1 that is subject to auto-regulatory feedback. Blood flow h 2 responds in proportion to this signal and caus... |

120 |
Statistical Mechanics
- Feynman
- 1972
(Show Context)
Citation Context ...ayes and ensemble learning This section reprises Friston et al. (2008), with a special focus on ensemble dynamics that form the basis of variational filtering. Variational Bayes or ensemble learning (=-=Feynman, 1972-=-; Hinton and von Cramp, 1993; MacKay, 1995; Attias, 2000) is a generic approach to model inversion that approximates the conditional density p(ϑ|y,m) on some model parameters, ϑ, given a model m and d... |

98 | Classical and Bayesian inference in neuroimaging: theory
- Friston, Penny, et al.
- 2002
(Show Context)
Citation Context ...nce and estimation procedures ranging from mixed-effects analyses in classical covariance component analysis to automatic relevance detection in machine learning formulations of related problems (see =-=Friston et al., 2002-=-, 2007 for a fuller discussion of hierarchical models of static data). The hierarchical forms for the states and predictions are 2 v 4 v 1 ð Þ v v m ð Þ 3 2 5 x 4 x 1 ðÞ v x m ð Þ 3 f x 5 f 1 v ... |

90 | The variational Bayesian EM algorithm for incomplete data: With application to scoring graphical model structures - Beal, Ghahramani - 2003 |

79 | Comparing dynamic causal models
- Penny, Stephan, et al.
(Show Context)
Citation Context ...the log-evidence. The objective is to compute q(ϑ) for each model by maximising the free-energy and then use F≈ln p(y|m) as a lower-bound approximation to the log-evidence for model comparison (e.g., =-=Penny et al., 2004-=-) or averaging (e.g., Trujillo-Barreto et al., 2004). Maximising the free-energy minimises the divergence, rendering the variational density q(ϑ)≈p(ϑ|y,m) an approximate posterior, which is exact for ... |

65 |
Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models
- Kass, Steffey
- 1989
(Show Context)
Citation Context ...pendencies, is shown in Fig. 2 (right panel). The conditional independence of the fluctuations induces a Markov property over levels, which simplifies the architecture of attending inference schemes (=-=Kass and Steffey, 1989-=-). A key property of hierarchical models is their connection to parametric empirical Bayes (Efron and Morris, 1973): Consider the energy function implied by model above Uu; ð tj# Þ lnp f yju 1 ð Þ ;... |

60 | Variational free energy and the Laplace approximation - Friston, Mattout - 2007 |

50 | Bayesian estimation of dynamical systems: an application to fMRI
- Friston
- 2002
(Show Context)
Citation Context ...n et al., 2004; Riera et al., 2004; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communications (Buxton et al., 1998; =-=Friston, 2002-=-). In brief, neuronal activity causes an increase in a vasodilatory signal h 1 that is subject to auto-regulatory feedback. Blood flow h 2 responds in proportion to this signal and causes changes in b... |

43 | The Unscented Particle Filter - Merwe, Doucety, et al. - 2000 |

39 |
Fluctuations and irreversible processes
- Onsager, Machlup
- 1953
(Show Context)
Citation Context ...tional formulations of optimal estimators for nonlinear state-space models (Eyink, 1996). Under linear dynamics, the effective action coincides with the Onsager–Machlup action in statistical physics (=-=Onsager and Machlup, 1953-=-; Graham, 1978). The action represents a lower-bound on the integral of logevidence over time, which, in the context of uncorrelated noise, is simply the log-evidence of the time-series. We now seek q... |

38 |
Stein’s estimation rule and its competitors – an empirical Bayes approach
- Efron, Morris
- 1973
(Show Context)
Citation Context ...dditive noise. This model has many conventional models as special cases. Critically, it is formulated in generalised coordinates, such that the evolution of the states is subject to empirical priors (=-=Efron and Morris, 1973-=-). This makes the states accountable to their conditional velocity through empirical priors on the dynamics (similarly for high-order motion). Special cases of this generalised model include state-spa... |

33 | Modeling the hemodynamic response to brain activation." Neuroimage 2004; 23 Suppl 1
- Buxton, Uludag, et al.
(Show Context)
Citation Context ...inference about brain states from non-invasive neurophysiologic observations is an important issue in cognitive neuroscience and functional imaging (e.g., Friston et al., 2003; Gitelman et al., 2003; =-=Buxton et al., 2004-=-; Riera et al., 2004; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communications (Buxton et al., 1998; Friston, 2002)... |

33 | A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals
- Riera, Watanabe, et al.
- 2004
(Show Context)
Citation Context ... states from non-invasive neurophysiologic observations is an important issue in cognitive neuroscience and functional imaging (e.g., Friston et al., 2003; Gitelman et al., 2003; Buxton et al., 2004; =-=Riera et al., 2004-=-; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communications (Buxton et al., 1998; Friston, 2002). In brief, neuronal... |

31 |
Modeling regional and psychophysiologic interactions in fMRI: the importance of hemodynamic deconvolution
- Gitelman, Penny, et al.
- 2003
(Show Context)
Citation Context ...as been chosen because inference about brain states from non-invasive neurophysiologic observations is an important issue in cognitive neuroscience and functional imaging (e.g., Friston et al., 2003; =-=Gitelman et al., 2003-=-; Buxton et al., 2004; Riera et al., 2004; Sotero and Trujillo-Barreto, in press). The hemodynamic model The hemodynamic model has been described extensively in previous communications (Buxton et al.,... |

26 |
Path-integral methods in nonequilibrium thermodynamics and statistics
- Graham
- 1978
(Show Context)
Citation Context ...mal estimators for nonlinear state-space models (Eyink, 1996). Under linear dynamics, the effective action coincides with the Onsager–Machlup action in statistical physics (Onsager and Machlup, 1953; =-=Graham, 1978-=-). The action represents a lower-bound on the integral of logevidence over time, which, in the context of uncorrelated noise, is simply the log-evidence of the time-series. We now seek q(u,t) which ma... |

22 | Variational mixture of Bayesian independent component analyzers
- Choudrey, Roberts
- 2003
(Show Context)
Citation Context ... they cover most parametric models one could conceive of; from independent component analysis to generalised convolution models. The relationship among these special cases is itself a large area (see =-=Choudrey and Roberts, 2001-=-), to which we will devote a subsequent paper. Here, we simply describe the general form of these models and their inversion. Hierarchical 5 When the states have a Markov blanket (i.e., there are unkn... |

19 |
A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach
- Ozaki
- 1992
(Show Context)
Citation Context ... that changes in the energy gradients are accommodated properly in the integration scheme. There are several ways to integrate these equations; we use a computationally intensive but accurate scheme (=-=Ozaki, 1992-=-) based on the matrix exponential of the system’s Jacobian, I(t). Ozaki (1992) shows the ensuing updates are consistent, coincide with the true trajectory (at least for linear systems) and retain the ... |

18 | On unscented Kalman filtering for state estimation of continuous-time nonlinear systems - Sarkka - 2007 |

12 | A mean field approximation in data assimilation for non-linear dynamics - Eyink, Restrepo, et al. - 2004 |

11 |
Free energy minimisation algorithm for decoding and cryptoanalysis
- MacKay
- 1995
(Show Context)
Citation Context ...rises Friston et al. (2008), with a special focus on ensemble dynamics that form the basis of variational filtering. Variational Bayes or ensemble learning (Feynman, 1972; Hinton and von Cramp, 1993; =-=MacKay, 1995-=-; Attias, 2000) is a generic approach to model inversion that approximates the conditional density p(ϑ|y,m) on some model parameters, ϑ, given a model m and data y. We will call the approximating cond... |

10 | Action principle in nonequilibrium statistical dynamics
- Eyink
- 1996
(Show Context)
Citation Context ...imators in time-series analysis. When q(u,t) shrinks to a point estimator, action reduces to the ‘effective action’ in variational formulations of optimal estimators for nonlinear state-space models (=-=Eyink, 1996-=-). Under linear dynamics, the effective action coincides with the Onsager–Machlup action in statistical physics (Onsager and Machlup, 1953; Graham, 1978). The action represents a lower-bound on the in... |

9 | 2008. Biophysical model for integrating neuronal activity, EEG, fMRI and metabolism - Sotero, Trujillo-Barreto |

8 | Keeping neural networks simple by minimising the description length of weights - Hinton, Cramp - 1993 |

4 |
DEM: a variational treatment of dynamic systems. Neuroimage
- KJ, Trujillo-Barreto, et al.
- 2008
(Show Context)
Citation Context ...filtering; Generalised coordinates Introduction Recently, we introduced a generic scheme for inverting dynamic causal models of systems with random fluctuations on exogenous inputs and hidden states (=-=Friston et al., 2008-=-). This scheme was called dynamic expectation maximisation (DEM) and assumed that the conditional densities on the system's states and parameters were Gaussian. This assumption is know as the Laplace ... |

4 |
Bayesian model averaging
- Trujillo-Barreto, Aubert-Vazquez, et al.
- 2004
(Show Context)
Citation Context ...mpute q(ϑ) for each model by maximising the free-energy and then use F≈ln p(y|m) as a lower-bound approximation to the log-evidence for model comparison (e.g., Penny et al., 2004) or averaging (e.g., =-=Trujillo-Barreto et al., 2004-=-). Maximising the free-energy minimises the divergence, rendering the variational density q(ϑ)≈p(ϑ|y,m) an approximate posterior, which is exact for simple (e.g., linear) systems. This can then be use... |

3 | A variational formulation of optimal nonlinear estimation - Eyink - 2001 |

3 | Generalized phase space version of Langevin equations and associated Fokker-Planck equations - Kerr, Graham |

3 | High-order variational perturbation theory for the free energy - Weissbach, Pelster, et al. - 2002 |

1 | This is relevant for Kalman filtering and related nonlinear Bayesian tracking schemes that assume wt − 1 is a well-behaved noise sequence. We have used the term process noise to distinguish it from system noise, w(t) in hierarchical dynamic models. This - Archambeau, Cornford, et al. - 2007 |

1 | A variational Bayesian framework for graphical models - Leen, T |