A Bayesian alternative to mutual information for the hierarchical clustering of dependent random variables

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering

A Theoretical Investigation of the Relationship between Structural Equation Modeling and Partial Correlation in Functional MRI Effective Connectivity

An important field of blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) is the investigation of effective connectivity, that is, the actions that a given set of regions exert on one another. We recently proposed a data-driven method based on the partial correlation matrix that could provide some insight regarding the pattern of functional interaction between brain regions as represented by structural equation modeling (SEM). So far, the efficiency of this approach was mostly based on empirical evidence. In this paper, we provide theoretical fundaments explaining why and in what measure structural equation modeling and partial correlations are related. This gives better insight regarding what parts of SEM can be retrieved by partial correlation analysis and what remains inaccessible. We illustrate the different results with real data

Automated extraction of mutual independence patterns using Bayesian comparison of partition models

Mutual independence is a key concept in statistics that characterizes the structural relationships between variables. Existing methods to investigate mutual independence rely on the definition of two competing models, one being nested into the other and used to generate a null distribution for a statistic of interest, usually under the asymptotic assumption of large sample size. As such, these methods have a very restricted scope of application. In the present manuscript, we propose to change the investigation of mutual independence from a hypothesis-driven task that can only be applied in very specific cases to a blind and automated search within patterns of mutual independence. To this end, we treat the issue as one of model comparison that we solve in a Bayesian framework. We show the relationship between such an approach and existing methods in the case of multivariate normal distributions as well as cross-classified multinomial distributions. We propose a general Markov chain Monte Carlo (MCMC) algorithm to numerically approximate the posterior distribution on the space of all patterns of mutual independence. The relevance of the method is demonstrated on synthetic data as well as two real datasets, showing the unique insight provided by this approach.Comment: IEEE Transactions on Pattern Analysis and Machine Intelligence (in press

Estimation of the hemodynamic response in event-related functional MRI: Bayesian networks as a framework for efficient Bayesian modeling and inference.

International audienceA convenient way to analyze blood-oxygen-level-dependent functional magnetic resonance imaging data consists of modeling the whole brain as a stationary, linear system characterized by its transfer function: the hemodynamic response function (HRF). HRF estimation, though of the greatest interest, is still under investigation, for the problem is ill-conditioned. In this paper, we recall the most general Bayesian model for HRF estimation and show how it can beneficially be translated in terms of Bayesian graphical models, leading to 1) a clear and efficient representation of all structural and functional relationships entailed by the model, and 2) a straightforward numerical scheme to approximate the joint posterior distribution, allowing for estimation of the HRF, as well as all other model parameters. We finally apply this novel technique on both simulations and real data

Estimation of the Hemodynamic Response Function in event-related functional MRI: directed acyclic graphs for a general Bayesian inference framework.

International audienceA convenient way to analyze BOLD fMRI data consists of modeling the whole brain as a stationary, linear system characterized by its transfer function: the Hemodynamic Response Function (HRF). HRF estimation, though of the greatest interest, is still under investigation, for the problem is ill-conditioned. In this paper, we recall the most general Bayesian model for HRF estimation and show how it can beneficially be translated in terms of graphical models, leading to (i) a clear and efficient representation of all structural and functional relationships entailed by the model, and (ii) a straightforward numerical scheme to approximate the joint posterior distribution, allowing for estimation of the HRF, as well as all other model parameters. We finally apply this novel technique on both simulations and real data

Robust Bayesian estimation of the hemodynamic response function in event-related BOLD fMRI using basic physiological information.

International audienceIn BOLD fMRI data analysis, robust and accurate estimation of the Hemodynamic Response Function (HRF) is still under investigation. Parametric methods assume the shape of the HRF to be known and constant throughout the brain, whereas non-parametric methods mostly rely on artificially increasing the signal-to-noise ratio. We extend and develop a previously proposed method that makes use of basic yet relevant temporal information about the underlying physiological process of the brain BOLD response in order to infer the HRF in a Bayesian framework. A general hypothesis test is also proposed, allowing to take advantage of the knowledge gained regarding the HRF to perform activation detection. The performances of the method are then evaluated by simulation. Great improvement is shown compared to the Maximum-Likelihood estimate in terms of estimation error, variance, and bias. Robustness of the estimators with regard to the actual noise structure or level, as well as the stimulus sequence, is also proven. Lastly, fMRI data with an event-related paradigm are analyzed. As suspected, the regions selected from highly discriminating activation maps resulting from the method exhibit a certain inter-regional homogeneity in term of HRF shape, as well as noticeable inter-regional differences

Estimation régularisée et non supervisée de la réponse hémodynamique en imagerie cérébrale fonctionnelle (IRMf)

L'estimation de la fonction de réponse hémodynamique (FRH) en imagerie par résonance magnétique fonctionnelle (IRMf) est essentielle pour une meilleure compréhension des activations cérébrales. Nous abordons ce problème dans un cadre bayésien en introduisant un a priori temporel sur la FRH et sous une forme non supervisée en maximisant la log-vraisemblance vis-à-vis des hyperparamètres. L'originalité de ce travail réside dans la définition d'une nouvelle fonction de vraisemblance, au nombre de paramètres réduit, qui vise d'une part à améliorer la prise en compte de la variabilité des artefacts physiologiques (coeur, respiration), et d'autre part à accélérer la convergence de l'algorithme EM utilisé pour la maximiser. Nous montrons l'intérêt de cette approche en la comparant aux travaux pré-existants [1], à la fois en simulation et sur données réelles

Assessing the Influence of Different ROI Selection Strategies on Functional Connectivity Analyses of fMRI Data Acquired During Steady-State Conditions

In blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI), assessing functional connectivity between and within brain networks from datasets acquired during steady-state conditions has become increasingly common. However, in contrast to connectivity analyses based on task-evoked signal changes, selecting the optimal spatial location of the regions of interest (ROIs) whose timecourses will be extracted and used in subsequent analyses is not straightforward. Moreover, it is also unknown how different choices of the precise anatomical locations within given brain regions influence the estimates of functional connectivity under steady-state conditions. The objective of the present study was to assess the variability in estimates of functional connectivity induced by different anatomical choices of ROI locations for a given brain network. We here targeted the default mode network (DMN) sampled during both resting-state and a continuous verbal 2-back working memory task to compare four different methods to extract ROIs in terms of ROI features (spatial overlap, spatial functional heterogeneity), signal features (signal distribution, mean, variance, correlation) as well as strength of functional connectivity as a function of condition. We show that, while different ROI selection methods produced quantitatively different results, all tested ROI selection methods agreed on the final conclusion that functional connectivity within the DMN decreased during the continuous working memory task compared to rest

A Network Analysis Approach to fMRI Condition-Specific Functional Connectivity

In this work we focus on examination and comparison of whole-brain functional connectivity patterns measured with fMRI across experimental conditions. Direct examination and comparison of condition-specific matrices is challenging due to the large number of elements in a connectivity matrix. We present a framework that uses network analysis to describe condition-specific functional connectivity. Treating the brain as a complex system in terms of a network, we extract the most relevant connectivity information by partitioning each network into clusters representing functionally connected brain regions. Extracted clusters are used as features for predicting experimental condition in a new data set. The approach is illustrated on fMRI data examining functional connectivity patterns during processing of abstract and concrete concepts. Topological (brain regions) and functional (level of connectivity and information flow) systematic differences in the ROI-based functional networks were identified across participants for concrete and abstract concepts. These differences were sufficient for classification of previously unseen connectivity matrices as abstract or concrete based on training data derived from other people

Cumulants of multiinformation density in the case of a multivariate normal distribution

International audienceWe consider a generalization of information density to a partitioning into N ≥ 2 subvectors. We calculate its cumulant-generating function and its cumulants, showing that these quantities are only a function of all the regression coefficients associated with the partitionin