Table 1.

The unnormalized joint posterior distribution of the parameters given the data, x_i and y_i. Following work by Horton et al. (2017), we incorporate population level parameters for pulse mass, width, baseline and half-life as we are jointly modeling a population of subjects. x_i and y_i are the vector of hormone concentration values for subject i.

Model component	Hierarchical Specification
(a) Likelihood	${\prod }_{i=1}^{m}L({\mathit{x}}_i,{\mathit{y}}_i\mid {\mathit{\theta }}_{s,i}^x,{\theta }_{b,i}^x,{\theta }_{h,i}^x,{\mathit{\theta }}_{s,i}^y,{\theta }_{b,i}^y,{\theta }_{h,i}^y,{\mathrm{\sum }}_{e,i})\times$
(b) Joint prior pulse locations	${\prod }_{i=1}^m\begin{array}{|c|}\hline \pi ({\mathit{\tau }}_i^y\mid {\mathit{\tau }}_i^x,\rho ,\nu )\pi ({\mathit{\tau }}_i^x\mid \beta ,R,\gamma )\\ \hline\end{array}\pi (\rho )\pi (\nu )\times$
(c) Joint prior pulse Mass & width	${\prod }_{i=1}^m\left\{{\prod }_{k=1}^{N_{s,i}^x}\pi ({({\alpha }_{s,ik}^x,{\omega }_{s,ik}^x)}^{\prime }\mid N_{s,i}^x,{\mathit{\mu }}_{s,i}^x,{\mathrm{\Upsilon }}_s^x){\prod }_{l=1}^{N_{s,i}^y}\pi ({({\alpha }_{s,il}^y,{\omega }_{s,il}^y)}^{\prime }\mid N_{s,i}^y,{\mathit{\mu }}_{s,i}^y,{\mathrm{\Upsilon }}_s^y)\right\}\times$
	${\prod }_{i=1}^m\begin{array}{|c|}\hline \pi ({({\mathit{\mu }}_{s,i}^x,{\mathit{\mu }}_{s,i}^y)}^{\prime }\mid {\mathit{\mu }}_s^{x,y},{\mathrm{\sum }}_s^{x,y})\\ \hline\end{array}\pi ({\mathit{\mu }}_s^{x,y})\pi ({\mathrm{\sum }}_s^{x,y})\pi ({\mathrm{\Upsilon }}_s^x)\pi ({\mathrm{\Upsilon }}_s^y)\times$
(d) Priors baseline	${\prod }_{i=1}^m\pi ({\theta }_{b,i}^x\mid {\mu }_b^x,{\sigma }_{x,b}^2)\pi ({\mu }_b^x)\pi ({\sigma }_{x,b}^2){\prod }_{i=1}^N\pi ({\theta }_{b,i}^y\mid {\mu }_b^y,{\sigma }_{y,b}^2)\pi ({\mu }_b^y)\pi ({\sigma }_{y,b}^2)\times$
(e) Priors half-life	${\prod }_{i=1}^m\pi ({\theta }_{h,i}^x\mid {\mu }_h^x,{\sigma }_{x,h}^2)\pi ({\mu }_h^x)\pi ({\sigma }_{x,h}^2){\prod }_{i=1}^N\pi ({\theta }_{h,i}^y\mid {\mu }_h^y,{\sigma }_{y,h}^2)\pi ({\mu }_h^y)\pi ({\sigma }_{y,h}^2)\times$
(f) Priors model error	${\prod }_{i=1}^m\pi ({\mathrm{\sum }}_{e.i})$