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In functional magnetic resonance imaging (fMRI) data processing, second-level Bayesian inference refers to the application of Bayes factors (BFs) as indicators for second-level analysis/group analysis in examining regional brain activity. Bayesian inference has emerged as a competing alternative to p-values and frequentist explorations of type I and type II errors, primarily due to how Bayes Factors enable experimenters to measure statistical evidence given prior parameters and evidence rather than assuming a fixed pre-set of parameters. Due to the notoriety of high false positive activations during fMRI post-processing, leading techniques including Bonferroni correction and False Discovery Rates (FDRs) have been implemented to minimize type I errors.[1] However, recent reports suggest that these frequentist toolkits may be too liberal or harsh in controlling for type I errors, proposing random field theory (RFT) familywise error correction (FWE)-applied voxel-wise thresholding as an appropriate balance. Nevertheless, follow-up discussions suggest that even popular methods including RFT do not attain sufficient significance levels, instead inflating false positive rates.
Since Bayesian inference does not presume parameter values, or effect sizes, as precisely equal to a definite value, type I/II error interpretations no longer have utility. Instead, by applying Bayes factors, the effect size can be expressed as uncertainty in terms of a probability distribution based on a prior distribution, incoming data, and its updated posterior distribution on parameters. Ultimately, a Bayesian framework is free from presumptions about effects that are certainly zero or null hypotheses, rendering it less vulnerable to inflated false positive rates. In contrast to a P-value, which quantifies the probability that one will observe values of a test statistic that are as more or less extreme than observed results, a Bayesian framework permits researchers to express a posterior uncertainty of voxel activity hypotheses. The capacity for Bayes Factors as a tool to accept hypotheses, including null, based on a ratio of posterior probabilities between (null hypothesis) and allows scientists to more readily make a decision on how to accept the null of the alternative[2][3].