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. 2013 Aug;22(4):364-81.
doi: 10.1177/0962280212448970. Epub 2012 May 23.

A Bayesian Non-Parametric Potts Model With Application to Pre-Surgical FMRI Data

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Free PMC article

A Bayesian Non-Parametric Potts Model With Application to Pre-Surgical FMRI Data

Timothy D Johnson et al. Stat Methods Med Res. .
Free PMC article

Abstract

The Potts model has enjoyed much success as a prior model for image segmentation. Given the individual classes in the model, the data are typically modeled as Gaussian random variates or as random variates from some other parametric distribution. In this article, we present a non-parametric Potts model and apply it to a functional magnetic resonance imaging study for the pre-surgical assessment of peritumoral brain activation. In our model, we assume that the Z-score image from a patient can be segmented into activated, deactivated, and null classes, or states. Conditional on the class, or state, the Z-scores are assumed to come from some generic distribution which we model non-parametrically using a mixture of Dirichlet process priors within the Bayesian framework. The posterior distribution of the model parameters is estimated with a Markov chain Monte Carlo algorithm, and Bayesian decision theory is used to make the final classifications. Our Potts prior model includes two parameters, the standard spatial regularization parameter and a parameter that can be interpreted as the a priori probability that each voxel belongs to the null, or background state, conditional on the lack of spatial regularization. We assume that both of these parameters are unknown, and jointly estimate them along with other model parameters. We show through simulation studies that our model performs on par, in terms of posterior expected loss, with parametric Potts models when the parametric model is correctly specified and outperforms parametric models when the parametric model in misspecified.

Keywords: Dirichlet process; FMRI; Potts model; decision theory; hidden Markov random field; non-parametric Bayes.

Figures

Figure 1
Figure 1
Results from the NP-Potts model. Top row: Four sagittal slices of the high resolution FLAIR image in the left hemisphere. Second row: FLAIR image with activation overlay. Bottom row: FLAIR image with deactivation overlay. The gray-scale bar (color bar in electronic version) represents the posterior probability of both activation (for the second row) and deactivation (for the bottom row). The overlays are highly pixelated as the analysis is performed in the functional MRI space which has a much lower resolution than the high resolution FLAIR image. The hyperprior distribution on pi0 is pi0 ~ (Beta(0.95(0.2N), 0.05(0.2N)). Loss function is (12) with c1 = c2 = 4.
Figure 2
Figure 2
Results from the normal-gamma model. Top row: Four sagittal slices of the high resolution FLAIR image in the left hemisphere. Second row: FLAIR image with activation overlay. Bottom row: FLAIR image with deactivation overlay. The gray-scale bar (color bar in electronic version) represents the posterior probability of both activation (for the second row) and deactivation (for the bottom row). The overlays are highly pixelated as the analysis is performed in the functional MRI space which has a much lower resolution than the high resolution FLAIR image. The prior on pi0 is fixed: pi0 = 1/3. Loss function is (12) with c1 = c2 = 4.
Figure 3
Figure 3
Histograms of the Z-scores from the final classifications into deactivated, null and activated states. The top histogram is from the DP-Potts model and the bottom histogram is from the normal-gamma model. The black histogram corresponds to the null classified voxels and their Z-scores, The outlined histograms to the left of the null histogram correspond to the deactivated voxels and those to the right, the activated voxels.
Figure 4
Figure 4
Robustness of the NP-Potts model. The NP-Potts model performs on par with the correctly specified model and performs better than misspecified models. β0 = 0.25, β1 = ln(2) (marginal prior probability = 0.5). The gray scale bars represent the posterior expected loss from the model used to fit the data (see legend) generated from the true model.

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