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Bayesian Estimation of Conditional Independence Graphs Improves Functional Connectivity Estimates.

Hinne M, Janssen RJ, Heskes T, van Gerven MA - PLoS Comput. Biol. (2015)

Bottom Line: A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections.As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results.The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

View Article: PubMed Central - PubMed

Affiliation: Radboud University, Institute for Computing and Information Sciences, Nijmegen, the Netherlands.

ABSTRACT
Functional connectivity concerns the correlated activity between neuronal populations in spatially segregated regions of the brain, which may be studied using functional magnetic resonance imaging (fMRI). This coupled activity is conveniently expressed using covariance, but this measure fails to distinguish between direct and indirect effects. A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections. Importantly, provided the data are normally distributed, if two variables are conditionally independent given all other variables, their respective partial correlation is zero. In this paper, we propose a probabilistic generative model that allows us to estimate functional connectivity in terms of both partial correlations and a graph representing conditional independencies. Simulation results show that this methodology is able to outperform the graphical LASSO, which is the de facto standard for estimating partial correlations. Furthermore, we apply the model to estimate functional connectivity for twenty subjects using resting-state fMRI data. Results show that our model provides a richer representation of functional connectivity as compared to considering partial correlations alone. Finally, we demonstrate how our approach can be extended in several ways, for instance to achieve data fusion by informing the conditional independence graph with data from probabilistic tractography. As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results. This reveals that while we are able to infer a clear backbone of connectivity in our empirical results, the data are not accurately described by simply looking at the mode of the distribution over connectivity. The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

No MeSH data available.


A The generative model for the conditional dependencies graph and precision matrix. B The generative model for structural connectivity and the precision matrix, based on both BOLD time series X and probabilistic streamline counts N. Latent variables, observed variables and hyperparameters are indicated in white, yellow and grey, respectively.
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pcbi.1004534.g001: A The generative model for the conditional dependencies graph and precision matrix. B The generative model for structural connectivity and the precision matrix, based on both BOLD time series X and probabilistic streamline counts N. Latent variables, observed variables and hyperparameters are indicated in white, yellow and grey, respectively.

Mentions: The preliminaries described above allow us to specify the distribution that is central to this work, i.e. the joint posterior over both the conditional independence graph and the precision matrix (an illustration of the graphical model is provided in Fig 1A):P(G,K∣X)∝P(X∣K)P(K∣G)P(G).(6)Note that the necessary hyperparameters are typically omitted for clarity.


Bayesian Estimation of Conditional Independence Graphs Improves Functional Connectivity Estimates.

Hinne M, Janssen RJ, Heskes T, van Gerven MA - PLoS Comput. Biol. (2015)

A The generative model for the conditional dependencies graph and precision matrix. B The generative model for structural connectivity and the precision matrix, based on both BOLD time series X and probabilistic streamline counts N. Latent variables, observed variables and hyperparameters are indicated in white, yellow and grey, respectively.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4634993&req=5

pcbi.1004534.g001: A The generative model for the conditional dependencies graph and precision matrix. B The generative model for structural connectivity and the precision matrix, based on both BOLD time series X and probabilistic streamline counts N. Latent variables, observed variables and hyperparameters are indicated in white, yellow and grey, respectively.
Mentions: The preliminaries described above allow us to specify the distribution that is central to this work, i.e. the joint posterior over both the conditional independence graph and the precision matrix (an illustration of the graphical model is provided in Fig 1A):P(G,K∣X)∝P(X∣K)P(K∣G)P(G).(6)Note that the necessary hyperparameters are typically omitted for clarity.

Bottom Line: A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections.As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results.The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

View Article: PubMed Central - PubMed

Affiliation: Radboud University, Institute for Computing and Information Sciences, Nijmegen, the Netherlands.

ABSTRACT
Functional connectivity concerns the correlated activity between neuronal populations in spatially segregated regions of the brain, which may be studied using functional magnetic resonance imaging (fMRI). This coupled activity is conveniently expressed using covariance, but this measure fails to distinguish between direct and indirect effects. A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections. Importantly, provided the data are normally distributed, if two variables are conditionally independent given all other variables, their respective partial correlation is zero. In this paper, we propose a probabilistic generative model that allows us to estimate functional connectivity in terms of both partial correlations and a graph representing conditional independencies. Simulation results show that this methodology is able to outperform the graphical LASSO, which is the de facto standard for estimating partial correlations. Furthermore, we apply the model to estimate functional connectivity for twenty subjects using resting-state fMRI data. Results show that our model provides a richer representation of functional connectivity as compared to considering partial correlations alone. Finally, we demonstrate how our approach can be extended in several ways, for instance to achieve data fusion by informing the conditional independence graph with data from probabilistic tractography. As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results. This reveals that while we are able to infer a clear backbone of connectivity in our empirical results, the data are not accurately described by simply looking at the mode of the distribution over connectivity. The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

No MeSH data available.