Abstract
Variational Bayes (VB), variational maximum likelihood (VML), restricted maximum likelihood (ReML), and maximum likelihood (ML) are cornerstone parametric statistical estimation techniques in the analysis of functional neuroimaging data. However, the theoretical underpinnings of these model parameter estimation techniques are rarely covered in introductory statistical texts. Because of the widespread practical use of VB, VML, ReML, and ML in the neuroimaging community, we reasoned that a theoretical treatment of their relationships and their application in a basic modeling scenario may be helpful for both neuroimaging novices and practitioners alike. In this technical study, we thus revisit the conceptual and formal underpinnings of VB, VML, ReML, and ML and provide a detailed account of their mathematical relationships and implementational details. We further apply VB, VML, ReML, and ML to the general linear model (GLM) with non-spherical error covariance as commonly encountered in the first-level analysis of fMRI data. To this end, we explicitly derive the corresponding free energy objective functions and ensuing iterative algorithms. Finally, in the applied part of our study, we evaluate the parameter and model recovery properties of VB, VML, ReML, and ML, first in an exemplary setting and then in the analysis of experimental fMRI data acquired from a single participant under visual stimulation.
Highlights
Variational Bayes (VB), variational maximum likelihood (VML), restricted maximum likelihood (ReML), and maximum likelihood (ML) are cornerstone parametric statistical estimation techniques in the analysis of functional neuroimaging data
Because VB, VML, ReML, and ML determine the scientific inferences drawn from empirical data in any of the above mentioned modeling frameworks, they are of immense importance for the neuroimaging practitioner
Because of their widespread use within the neuroimaging community, we reasoned that a theoretical treatment of these techniques in a familiar model scenario may be helpful for both neuroimaging novices, who would like to learn about some of the standard statistical estimation techniques employed in the field, and for neuroimaging practitioners, who would like to further explore the foundations of these and alternative model estimation approaches
Summary
Variational Bayes (VB), variational maximum likelihood (VML) ( known as expectationmaximization), restricted maximum likelihood (ReML), and maximum likelihood (ML) are cornerstone parametric statistical estimation techniques in the analysis of functional neuroimaging data. Because of their widespread use within the neuroimaging community, we reasoned that a theoretical treatment of these techniques in a familiar model scenario may be helpful for both neuroimaging novices, who would like to learn about some of the standard statistical estimation techniques employed in the field, and for neuroimaging practitioners, who would like to further explore the foundations of these and alternative model estimation approaches In this technical study, we revisit the conceptual underpinnings of the aforementioned techniques and provide a detailed account of their mathematical relations and implementational details. We make explicit reference to these functions, which share the stub vbg_∗.m
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