Elsevier

NeuroImage

Volume 45, Issue 1, Supplement 1, March 2009, Pages S173-S186
NeuroImage

Bayesian analysis of neuroimaging data in FSL

https://doi.org/10.1016/j.neuroimage.2008.10.055Get rights and content

Abstract

Typically in neuroimaging we are looking to extract some pertinent information from imperfect, noisy images of the brain. This might be the inference of percent changes in blood flow in perfusion FMRI data, segmentation of subcortical structures from structural MRI, or inference of the probability of an anatomical connection between an area of cortex and a subthalamic nucleus using diffusion MRI. In this article we will describe how Bayesian techniques have made a significant impact in tackling problems such as these, particularly in regards to the analysis tools in the FMRIB Software Library (FSL). We shall see how Bayes provides a framework within which we can attempt to infer on models of neuroimaging data, while allowing us to incorporate our prior belief about the brain and the neuroimaging equipment in the form of biophysically informed or regularising priors. It allows us to extract probabilistic information from the data, and to probabilistically combine information from multiple modalities. Bayes can also be used to not only compare and select between models of different complexity, but also to infer on data using committees of models. Finally, we mention some analysis scenarios where Bayesian methods are impractical, and briefly discuss some practical approaches that we have taken in these cases.

Introduction

In a typical neuroimaging scenario we are looking to extract pertinent information about the brain from noisy data. Mapping measured data to brain characteristics is generally difficult to do directly. For example, since FMRI data is noisy we cannot simply use a rule that says “If the FMRI data looks exactly like X, then the brain is active in area Y”. However, it is comparatively easy to turn the problem around and specify “If the brain is active in area Y, then the FMRI data should look like X”, i.e., if we know what the brain is doing then we can predict what our neuroimaging data should look like. This is what we refer to as a Generative Model, which sits at the heart of all Bayesian neuroimaging analysis methods. Fig. 1 illustrates an example of a generative model for predicting FMRI data.

Generative models are a natural way for us to incorporate our understanding of the brain and of the neuroimaging modalities to make predictions about what neuroimaging data looks like. However, in practice we want to do the opposite. We want to be able to take acquired data (plus a generative model) and extract pertinent information about the brain (i.e., “infer” on the model and its parameters). The classical approach to doing this is to fit the generative models to the data, for example by minimising the squared difference between the data and the generative model to estimate each parameter in the model1. However, this approach has limitations. Firstly, extracting a single “best guess” (or point estimate) for a parameter completely ignores the presence of, or extent of, the uncertainty that we have in that parameter. Secondly, how do we systematically combine the information in the data with any prior knowledge that we have about the parameters in the model? Bayesian statistics offers a solution to these problems, and as we shall see also provides a framework in which we can do much more besides. For example, we can probabilistically combine information from multiple modalities, compare and select between models of different complexity, and infer on data using committees of models.

Section snippets

Bayes' rule

Bayesian statistics provide the only generic framework for the adjustment of belief (in the form of probability density functions (PDFs)) in the presence of new information (Cox, 1946). They give us a tool for inferring on any model we choose, and guarantee that uncertainty will be handled correctly.

Bayes' rule tells us how (for a model M) we should use the data, Y, to update our prior belief in the values of the parameters Θ, p(Θ|M) to a posterior distribution of the parameter values p(Θ|Y,M):p

Priors

Bayesian statistics requires that we specify our prior probabilistic belief about the model parameters. This requirement has often been a source of criticism of the Bayesian approach. However, Bayesians support the view that we cannot infer from data without making assumptions; indeed, the act of choosing a generative model itself constitutes an assumption (that the model provides a good description of reality). It turns out that having a framework within which we can specify prior assumptions

Bayesian versus frequentist inference

Until recently nearly all inference on functional neuroimaging data (e.g., PET, FMRI) was carried out in a frequentist framework; only in the last five years or so has Bayes started to be used either alongside or to replace frequentist inference. In classical frequentist statistics, probabilities refer to the frequency of outcomes. In contrast, Bayesians use probabilities to express degrees of belief. Care should be taken to not confuse P-values with Bayesian probabilities: Bayesian methods can

Multi-modality fusion

In neuroimaging we can often obtain complementary information about some key underlying physiological phenomenon from more than one imaging modality. A key question is: how do we go about optimally combining this information? For example, if information extracted from a T1-weighted structural MR image tells us that a particular voxel is white matter, whereas a proton density MR structural tells us that the same voxel is grey matter, then what should we infer? The answer is to use Bayes.

As

Hierarchical models — combining local and global information

In the Bayes' rule section, we saw how spatial priors can be used to spatially regularise model parameters, in other words, how to model dependency between neighbouring imaging voxels. Bayesian techniques can also be used to allow data from remote voxels to depend on each other, via the use of hierarchical priors. Inference made on local parameters specific to one voxel, Θi, may then be influenced by inference made in other voxels, if the parameters from these voxels share a hierarchical prior.

Model selection

Bayesian data analysis is the process of fitting a probabilistic model to a set of data, and encompasses the following three main steps: (i) setting up a full probabilistic model, including a data generative model (likelihood function) and a set of priors on model parameters; (ii) conditioning on observed data (posterior distribution); and (iii) evaluating the performance of the model. The last step is one of the strengths of Bayesian techniques. Given that any model used within a Bayesian

Model averaging

When performing Bayesian model selection, inference on the parameters is made for each model separately, and the evidences may subsequently be compared to select the appropriate model. An alternative method is Bayesian model averaging, where inference is derived from a committee of models. Posterior PDFs on model parameters are then a weighted average of the posteriors given each model, where the weights are the respective evidences of each model:p(Θ|Y,{M1,,Mn})=k=1np(Θ|Y,Mk)p(Mk|Y).While

Bayesian inference techniques

Throughout this article we have considered some of the key concepts in Bayes and how they can be used to our advantage in performing neuroimage analysis. However, we have largely ignored one of the key difficulties in using Bayes: that is that the solution to Bayes' equation is seldom straightforward. In particular, the integrals in Eqs. (3) and (4) are not in general analytically tractable (i.e., we cannot solve them mathematically). As illustrated in Fig. 14, we briefly consider here some of

Non-Bayesian approaches

We have described the advantages in the Bayesian approach of estimating the full posterior distribution of model parameters of interest, as opposed to just obtaining a point estimate, or “best guess”, of each model parameter, such as one achieves with maximum likelihood approaches. However, there are many scenarios in the analysis of neuroimaging data where the Bayesian approach is impractical. Typically, this occurs when the model and the interaction between model parameters become too complex

Conclusion

In this article we have described the impact that Bayes has had on neuroimaging in the research experiences of the authors. In particular we have considered Bayesian concepts such as biophysical and regularisation priors, hierarchical modelling, and model selection and averaging. Bayes provides us with a framework which (computational practicalities aside) allows us to infer on any generative model that we can propose. As computational resources increase then so will the quality of information

Role of the funding source

Mark Woolrich was funded by the UK EPSRC, Saad Jbabdi by the Hadwen Trust, Michael Chappell by GlaxoSmithKline, Salima Makni by the UK EPSRC, Timothy Behrens by the UK MRC, and Mark Jenkinson by the UK BBSRC.

Conflict of interest statement

The authors declare that they have no conflict of interest relative to this research.

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