Introduction
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Related work
Methods
Study protocol
Physiological measurements
Data preparation
Hierarchical Bayesian models for location and scale
Hierarchical Bayesian models for the mean of a Gaussian distribution are conceptually the Bayesian equivalent of the ANOVA \cite{Kruschke2015}. However, the Bayesian framework has additional flexibility that may be used to relax some of the traditional assumptions of ANOVA. Firstly, within the Bayesian framework it is straightforward to model the scale of the distribution as well, thus relaxing the ANOVA assumption of equal variances between groups \cite{Kruschke2015}. Secondly, we are free to choose the sampling distribution for the modeled data from any distribution supported by the probabilistic programming software in use. In other words, other distributions than the Gaussian may be used, for example, the fatter tailed Cauchy to accommodate for outliers . In addition to these, the use of Bayesian statistics brings on the usual benefits of probabilistic models: questions such as "Does x differ from y?" can be intuitively and effectively answered using the posterior, computing probabilities for statements like "What is the probability that the true value of alpha is larger than xx?" is straightforward and the model can be used to make predictions: a point estimates with quantified uncertainty.
To analyze the IBI data we used the following base model: