Classical and Bayesian Inference in Neuroimaging: Applications

Abstract

In Friston et al. ((2002)Neuroimage 16: 465-483) we introduced empirical Bayes as a potentially useful way to estimate and make inferences about effects in hierarchical models. In this paper we present a series of models that exemplify the diversity of problems that can be addressed within this framework. In hierarchical linear observation models, both classical and empirical Bayesian approaches can be framed in terms of covariance component estimation (e.g., variance partitioning). To illustrate the use of the expectationmaximization (EM) algorithm in covariance component estimation we focus first on two important problems in fMRI: nonsphericity induced by (i) serial or temporal correlations among errors and (ii) variance components caused by the hierarchical nature of multisubject studies. In hierarchical observation models, variance components at higher levels can be used as constraints on the parameter estimates of lower levels. This enables the use of parametric empirical Bayesian (PEB) estimators, as distinct from classical maximum likelihood (ML) estimates. We develop this distinction to address: (i) The difference between response estimates based on ML and the conditional means from a Bayesian approach and the implications for estimates of intersubject variability. (ii) The relationship between fixed-and random-effect analyses. (iii) The specificity and sensitivity of Bayesian inference and, finally, (iv) the relative importance of the number of scans and subjects. The forgoing is concerned with within-and between-subject variability in multisubject hierarchical fMRI studies. In the second half of this paper we turn to Bayesian inference at the first (within-voxel) level, using PET data to show how priors can be derived from the (between-voxel) distribution of activations over the brain. This application uses exactly the same ideas and formalism but, in this instance, the second level is provided by observations over voxels as opposed to subjects. The ensuing posterior probability maps (PPMs) have enhanced anatomical precision and greater face validity, in relation to underlying anatomy. Furthermore, in comparison to conventional SPMs they are not confounded by the multiple comparison problem that, in a classical context, dictates high thresholds and low sensitivity. We conclude with some general comments on Bayesian approaches to image analysis and on some unresolved issues. © 2002 Elsevier Science (USA)

Immediate memory for “when, where and what”: Short‐delay retrieval using dynamic naturalistic material

“…These subject‐specific contrast images were entered in the second‐level ANOVA. For the identification of task‐specific retrieval effects, we compared each task with the mean of the two other tasks, averaging the two Temporal conditions: “T (short+long) /2 > (S + O)/2”; “S > (T (short+long) /2 + O)/2”; and “O > (T (short+long) /2 + S)/2.” Sphericity‐correction was applied to all group‐level ANOVAs to account for any nonindependent error term for repeated measures and any difference in error variance across conditions [Friston et al, 2002]. The P ‐values were corrected for multiple comparisons using a cluster‐level threshold of P‐ FWE‐corr.…”

Bayesian fMRI time series analysis with spatial priors

“…Previous use of spatial priors in fMRI has either relied on the use of hard (i.e., non-adaptive) priors and projection of the data onto a subspace spanned by these priors (see, for example, Section 3 of (Friston et al, 2002)) or used adaptive priors but the computationally expensive MCMC algorithm to sample from the relevant posterior distributions (Gossl et al, 2001). In this paper, we have shown how to use a fast analytic approximation to the posterior distribution which was derived using the VB framework.…”

Accurate modeling of temporal correlations in rapidly sampled fMRI time series

“…As implemented in SPM (spm_reml.m), the algorithm used for the AR(1) + white noise model or the FAST model was exactly the same, with the exception that the number of hyperparameters was increased for the latter. The algorithm performs a Fisher scoring ascent on variational free energy (i.e., a lower bound on Bayesian model evidence) to identify maximum a posteriori covariance component (hyper) parameter estimates, as described in (Friston et al, 2002; Penny et al, 2007; Starke & Ostwald, 2017). The same priors as for the conventional AR(1) + white noise model were used for each hyperparameter, ${\mathrm{normal\lambda }}_i$.…”

“…The autocorrelation matrix V can be estimated, using Restricted Maximum Likelihood (ReML), as a linear combination of a fixed set of covariance components, $V={false\sum }_i{\mathrm{normal\lambda }}_i{C}^i$ modeling a mixture of white noise and a first‐order autoregressive process AR(1) (Friston et al, 2002). However, recent studies have shown that this simple model with two components may not be enough to model temporal correlations of accelerated sequences with more rapid sampling rates (Bollmann et al, 2018; Eklund et al, 2012; Olszowy et al, 2017).…”

“…Modeling temporal correlations is essential to preclude overestimation of functional sensitivity and to prevent an inflated false‐positive rate. A well‐established approach for modeling and removing temporal correlations is to use a mixture of white noise and a first‐order autoregressive model (Friston et al, 2002). However, with the advent of rapid imaging techniques, it has been shown that such a model may not sufficiently capture temporal correlations in time series acquired with very short TR (Bollmann, Puckett, Cunnington, & Barth, 2018; Eklund, Andersson, Josephson, Johannesson, & Knutsson, 2012; Olszowy, Williams, Rua, & Aston, 2017).…”