\( % Custom math symbols % new ones \newcommand{\Tau}{\mathrm{T}} % general \newcommand{\vect}[1]{\boldsymbol{#1}} % Vectors % Parameters - specific, frequently used \newcommand{\zetavec}{\vect{\zeta}} % vector of zetas existing pulses and characteristics %\newcommand{\xivec}{\vect{\xi}} % vector of xis \newcommand{\etavec}{\vect{\eta}} % vector of current pulses/locations \newcommand{\tauvec}{\vect{\tau}} % vector of current pulses/locations \newcommand{\taustar}{\tau^{*}} % tau* new pulse/location \newcommand{\tauprime}{\tau^{\prime}} % tau' specific existing pulse/location \newcommand{\omegavec}{\vect{\omega}} % vector of omega* current widths \newcommand{\omegastar}{\omega^{*}} % omega* new pulse width \newcommand{\omegaprime}{\omega^{\prime}} % omega' specific existing width \newcommand{\alphavec}{\vect{\alpha}} % vector of alpha existing masses \newcommand{\alphastar}{\alpha^{*}} % alpha* new pulse mass \newcommand{\alphaprime}{\alpha^{\prime}} % alpha' specific existing mass \newcommand{\mualpha}{\mu_{\alpha}} % mean pulse mass \newcommand{\muomega}{\mu_{\omega}} % mean pulse width \newcommand{\sigmaalpha}{\sigma_{\alpha}} % SD pulse mass \newcommand{\sigmaomega}{\sigma_{\omega}} % SD pulse width % Other parms \newcommand{\baseline}{\theta_{b}} % frequently used symbol alpha' \newcommand{\halflife}{\theta_{h}} % frequently used symbol alpha' \newcommand{\meanmass}{\mu_{\alpha}} % frequently used symbol alpha' \newcommand{\meanwidth}{\mu_{\omega}} % frequently used symbol alpha' \newcommand{\sdmass}{\sigma_{\alpha}} % frequently used symbol alpha' \newcommand{\sdwidth}{\sigma_{\omega}} % frequently used symbol alpha' \newcommand{\errorvar}{\sigma_{\epsilon}^{2}} % frequently used symbol alpha' \newcommand{\thetavec}{\vect{\theta}} % Equations - generalized, frequently used \newcommand{\srsum}[1]{\sum_{#1} S(R)} % S(R) with summation before \newcommand{\sr}[1]{S_{#1}(R)} % S(R) without summation \)

B Derivations of posterior distributions

B.1 Single-subject

B.1.1 \(\pi(\nu|\cdots)\) – Standard deviation of the random effects

$ \[\begin{align} \pi(\nu_{\alpha} | \cdots) &\propto \pi(\nu_{\alpha}) \pi(\alpha_i | \mu_{\alpha}, \nu_{\alpha}, \vect{\kappa}_{\alpha}) \\ where,\\ & \nu_{\alpha} \sim Unif(0, a)\\ & \alpha_i | \mu_{\alpha}, \nu_{\alpha}, \vect{\kappa}_{\alpha} \sim t^+_4 (\mu_{\alpha}, \nu^2_{\alpha})\\ \\ \end{align}\]

$

The distribution of \(\nu_{\alpha}\) is evident. The distribution of \(\pi(\alpha_i | \mu_{\alpha}, \nu_{\alpha}, \vect{\kappa}_{\alpha})\) is achieved using a normal-gamma mixture:

$ \[\begin{align} \pi(\alpha_i | \mu_{\alpha}, \nu_{\alpha}, \vect{\kappa}_{\alpha}) &\sim N^+ (\mu_{\alpha}, \frac{\nu^2_{\alpha}}{\vect{\kappa}_{\alpha}})\\ where,\\ & \vect{\kappa}_{\alpha} \sim \Gamma(r/2,r/2)\\ \end{align}\]

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