### Citations

12415 |
Elements of information theory
- Cover, Thomas
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Citation Context ... density. We have � p(Y, θ) F = q(θ|Y ) log dθ, (13) q(θ|Y ) which is known (to physicists) as the negative variational free energy and � q(θ|Y ) KL = q(θ|Y ) log dθ (14) p(θ|Y ) is the KL-divergence =-=[6]-=- between the density q(θ|Y ) and the true posterior p(θ|Y ). Equation 12 is the fundamental equation of the VB-framework. Importantly, because the KLdivergence is always positive [6], F provides a low... |

2193 | Bayesian Data Analysis
- Gelman, Carlin, et al.
- 2004
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Citation Context ...rior distribution p(θ|Y ). This implies estimation both of the parameters θ and the uncertainties associated with their estimation. This can be achieved using standard Markov Chain Monte Carlo (MCMC) =-=[12]-=- procedures to produce samples from the posterior. Brain imaging data sets are, however, 3 prohibitively large (typically N = 50, 000, T = 200) making MCMC impractical for routine use. We have therefo... |

1082 |
Statistical parametric maps in functional imaging: A general linear approach.
- riston, Holmes, et al.
- 1995
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Citation Context ...ferences about regionally specific activations in the human brain [7]. From measurements of changes in blood oxygenation one can use various statistical models, such as the General Linear Model (GLM) =-=[8]-=-, to make inferences about task-specific changes in underlying neuronal activity. In previous work [21, 23, 22] we have developed a spatially regularised General Linear Model (GLM) for the analysis of... |

746 |
Bayesian Inference in Statistical Analysis.
- Box, Tiao
- 1992
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Citation Context ...ontrast matrix C we have q(cn) = N(cn; mn, Vn) (32) 6 with mean and covariance mn = C ˆwn (33) Vn = C T ˆ ΣnC Bayesian inference based on this posterior can then take place using confidence intervals =-=[5]-=-. For univariate contrasts we have suggested the use of Posterior Probability Maps (PPMs). Before discussing this at length in section 4, we describe a new approach that allows us to make inferences a... |

329 | Nonlinear spatial normalization using basis functions.
- Ashburner, KJ
- 1999
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Citation Context ...red at different times, time series were interpolated to the acquisition time of the reference slice. Images were then spatially normalized to a standard EPI template using a nonlinear warping method =-=[2]-=-. Each time series was then high-pass filtered using a set of discrete cosine basis functions with a filter cut-off of 128 seconds. The data were then analysed using the design matrix shown in Figure ... |

192 | Human Brain Function.
- Frackowiak, Friston, et al.
- 1997
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Citation Context ...Functional Magnetic Resonance Imaging (fMRI) using Blood Oxygen Level Dependent (BOLD) contrast is an established method for making inferences about regionally specific activations in the human brain =-=[7]-=-. From measurements of changes in blood oxygenation one can use various statistical models, such as the General Linear Model (GLM) [8], to make inferences about task-specific changes in underlying neu... |

173 | Classical and Bayesian inference in neuroimaging: Theory.
- Friston, Penny, et al.
- 2002
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Citation Context ...oefficient estimated from the face fMRI data. In these images, black denotes 0 and white 1. Figure 11: PPM showing above threshold χ 2 statistics for any effect of facess6. Discussion In previous work=-=[10]-=-, we have compared the sensitivity and specificity of classical inference to Bayesian inference with Minimum Norm (MN) priors. This comparison was made possible by looking at the expected properties o... |

144 |
The Variational Bayesian EM Algorithm for Incomplete Data: with Application to Scoring Graphical Model Structures
- Beal, Ghahramani
- 2003
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Citation Context ... fine-tuning models. For example, the choice of hemodynamic basis set [22] or the order of the autoregressive models [21]. It is also possible to approximate the model evidence using sampling methods =-=[12, 4]-=-. In the very general context of probabilistic graphical models, Beal and Ghahramani [4] have shown that the VB approximation of model evidence is considerably more accurate than the Bayesian Informat... |

124 |
Numerical bayesian methods applied to signal processing,
- Ruanaidh, Fitzgerald
- 1996
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Citation Context ...he component posteriors (using equation 16). This is to be contrasted with Laplace approximations where we fix the form of the component posteriors to be Gaussian about the maximum posterior solution =-=[20]-=-. The above principles lead to a set of coupled update rules for the parameters of the component posteriors, iterated application of which leads to the desired maximisation. Further, by computing F fo... |

68 | Ensemble learning
- Lappalainen, Miskin
- 2000
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Citation Context ...e procedure which is computationally efficient. This allows inferences to be made within minutes rather than hours. The approach is called Variational Bayes (VB), a full tutorial on which is given in =-=[16]-=-. In what follows we describe the key features. Given a probabilistic model of the data, the log of the ‘evidence’ or ‘marginal likelihood’ can be written as � log p(Y ) = q(θ|Y ) log p(Y )dθ � p(Y, θ... |

66 | Bayesian fMRI time series analysis with spatial priors,”
- Penny, Trujillo-Barreto, et al.
- 2005
(Show Context)
Citation Context ... blood oxygenation one can use various statistical models, such as the General Linear Model (GLM) [8], to make inferences about task-specific changes in underlying neuronal activity. In previous work =-=[21, 23, 22]-=- we have developed a spatially regularised General Linear Model (GLM) for the analysis of fMRI data which allows for the characterisation of regionally specific effects using Posterior Probability Map... |

65 | Face repetition effects in implicit and explicit memory tests as measured by fMRI
- Henson, Shallice, et al.
- 2002
(Show Context)
Citation Context ...reshold voxels. The default thresholds were used, that is, we plot χ 2 n for voxels which satisfy p(cn > 0) > 1 − 1/N. 9 5.4 Face data This is an event-related fMRI data set acquired by Henson et al. =-=[15]-=-. The data were acquired during an experiment concerned with the processing of images of faces [15]. This was an event-related study in which greyscale images of faces were presented for 500ms, replac... |

56 | Posterior probability maps and SPMs.
- Friston, Penny
- 2003
(Show Context)
Citation Context ...re Dw is a spatial precision matrix. This can be set to correspond to eg. a Low Resolution Tomography (LORETA) prior, a Gaussian Markov Random Field (GMRF) prior or a Minimum Norm (MN) prior (Dw = I) =-=[9]-=- as described in earlier work [23]. These priors are specified separately for each slice of data. Specification of 3-dimensional spatial priors (ie. over multiple slices) is desirable from a modelling... |

55 | Variational Bayesian inference for fMRI time series,”
- Penny, Kiebel, et al.
- 2003
(Show Context)
Citation Context ... blood oxygenation one can use various statistical models, such as the General Linear Model (GLM) [8], to make inferences about task-specific changes in underlying neuronal activity. In previous work =-=[21, 23, 22]-=- we have developed a spatially regularised General Linear Model (GLM) for the analysis of fMRI data which allows for the characterisation of regionally specific effects using Posterior Probability Map... |

54 | Multivariate autoregressive modeling of fMRI time series.
- Harrison, Penny, et al.
- 2003
(Show Context)
Citation Context ...dent, squared Gaussian variables. As such it has a χ 2 distribution p(dn) = χ 2 (vn) (35) This procedure is identical to that used for making inferences in Bayesian Multivariate Autoregressive Models =-=[13]-=-. We can also use this procedure to test for two-sided effects, that is, activations or deactivations. Although these contrasts are univariate we will use the term ‘multivariate contrasts’ to also inc... |

52 | Constrained linear basis sets for HRF modelling using variational Bayes.
- Woolrich, Behrens, et al.
- 2004
(Show Context)
Citation Context ...his form of the prior is useful as our specification of approximate posteriors is based on similar quantities. The above Gaussian priors underly GMRFs and LORETA and have been used previously in fMRI =-=[26]-=- and EEG [18]. They are by no means, however, the optimal choice for imaging data. In EEG, for example, much interest has focussed on the use of L p -norm priors [3] instead of the L 2 -norm implicit ... |

46 | A new statistical approach to detecting significant activation in functional
- Marchini, Ripley
- 2000
(Show Context)
Citation Context ...Together, DCT and AR modelling can account for long-memory noise processes. Alternative procedures for removing low-frequency drifts include the use of running-line smoothers or polynomial expansions =-=[17]-=-. 2.1 Model likelihood We now describe the approach taken in our previous work. For a P th-order AR model, the likelihood of the data is given by p(Y |W, A, λ) = T� t=P +1 n=1 N� N(ytn − xtwn; (2) (dt... |

42 |
Mixture models with adaptive spatial regularization for segmentation with an application to fMRI data,”
- Woolrich, Behrens, et al.
- 2005
(Show Context)
Citation Context ...sitive rates for null fMRI data, and physiologically plausible activations for auditory and face fMRI data sets. A comprehensive Bayesian thresholding approach has been implemented by Woolrich et al. =-=[24]-=-. This work uses explicit models of the null and alternative hypotheses based on Gaussian and Gamma variates. This requires a further computationally expensive stage of model-fitting, based on spatial... |

24 | Bayesian Comparison of Spatially Regularised General Linear Models. Human Brain Mapping, 2006. Accepted for publication
- Penny, Flandin, et al.
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Citation Context ...ch is defined as less than a 1% increase in F , the objective function. For the empirical work in this paper, however, we fixed the number of iterations to 4. Expressions for computing F are given in =-=[22]-=-. This is an important quantity as it can also be used 5 for model comparison. This is described at length in [22]. The algorithm we have described is implemented in SPM version 5 and can be downloade... |

23 |
Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain.
- Pascal-Marqui, Michel, et al.
- 1994
(Show Context)
Citation Context ...he prior is useful as our specification of approximate posteriors is based on similar quantities. The above Gaussian priors underly GMRFs and LORETA and have been used previously in fMRI [26] and EEG =-=[18]-=-. They are by no means, however, the optimal choice for imaging data. In EEG, for example, much interest has focussed on the use of L p -norm priors [3] instead of the L 2 -norm implicit in the Gaussi... |

20 |
Bayesian analysis of the neuromagnetic inverse problem with ℓp-norm priors
- Auranen, Nummenmaa, et al.
- 2005
(Show Context)
Citation Context ...have been used previously in fMRI [26] and EEG [18]. They are by no means, however, the optimal choice for imaging data. In EEG, for example, much interest has focussed on the use of L p -norm priors =-=[3]-=- instead of the L 2 -norm implicit in the Gaussian assumption. Additionally, we are currently investigating the use of wavelet priors. This is an active area of research and will be the topic of futur... |

18 |
Temporal autocorrelation in univariate linear modelling of FMRI data
- Woolrich, Ripley, et al.
- 2001
(Show Context)
Citation Context ...rifts due to hardware instabilities, aliased cardiac pulsation and respiratory sources, unmodelled neuronal activity and residual motion artifacts not accounted for by rigid body registration methods =-=[25]-=-. This results in the residuals of an fMRI analysis being temporally autocorrelated. In previous work we have shown that, after removal of low-frequency drifts using Discrete Cosine Transform (DCT) ba... |

10 | Optimal spatial regularisation of autocorrelation estimates in fMRI analysis
- Gautama, Hulle
(Show Context)
Citation Context ...ets, low-order voxel-wise autoregressive (AR) models are sufficient for modelling this autocorrelation [21]. It is important to model these noise processes as parameter estimation becomes less biased =-=[11]-=- and more accurate [21]. Together, DCT and AR modelling can account for long-memory noise processes. Alternative procedures for removing low-frequency drifts include the use of running-line smoothers ... |

10 |
Analysis of fMRI time series
- Henson
- 2003
(Show Context)
Citation Context ...ental conditions, where each stimulus train has been convolved with two different hemodynamic bases (i) the canonical Hemodynamic Response Function (HRF) and (ii) the time derivative of the canonical =-=[14]-=-. The next 6 regressors in the design matrix describe movement of the subject in the scanner and the final column models the mean response. The model was then fitted using the VB algorithm. Figure 10 ... |

8 |
Bayesian analysis of single-subject fMRI: SPM implementation
- Penny, Flandin
- 2005
(Show Context)
Citation Context ... downloaded from [1]. Computation of a number of quantites (eg. ˜ Cn, ˜ dn and ˜ Gn) is now much more efficient than in previous versions [23]. These improvements are described in a separate document =-=[27]-=-. To analyse a single session of data (eg. the face fMRI data) takes about 30 minutes on a typical modern PC. 2.6 Spatio-temporal deconvolution The central quantity of interest in fMRI analysis is our... |

4 |
Wellcome Department of Imaging Neuroscience. Available at http://www.fil.ion.ucl.ac.uk/spm/software
- SPM
- 2002
(Show Context)
Citation Context ...is an important quantity as it can also be used 5 for model comparison. This is described at length in [22]. The algorithm we have described is implemented in SPM version 5 and can be downloaded from =-=[1]-=-. Computation of a number of quantites (eg. ˜ Cn, ˜ dn and ˜ Gn) is now much more efficient than in previous versions [23]. These improvements are described in a separate document [27]. To analyse a s... |

3 |
Algorithms for Approximate Bayesian Inference
- Variational
- 2003
(Show Context)
Citation Context ...B is of comparable accuracy to a much more computationally demanding method based on Annealed Importance Sampling (AIS) [4]. 2.4 Approximate Posteriors This paper uses the Variational Bayes framework =-=[19]-=- for estimation and inference. We describe the algorithm developed in previous work [23] in which we assumed that the approximate posterior factorises over voxels and subsets of parameters. Because of... |