Multi-fate nest survival

\[\begin{align} \left[ \boldsymbol{\beta}, \boldsymbol{\gamma}, \boldsymbol{\psi}, \boldsymbol{\sigma}, \textbf{z} \mid \textbf{Y} \right] \ \propto \ &\prod_{n=1}^N \prod_{t=1}^{T-1} \text{Categorical}\left(y_{nt+1} \mid \boldsymbol{\theta}_{nt} \right) \times \prod_{o=2}^O \prod_{k=1}^K \text{Student's }t\left({\beta_o}_k \mid 3, 0, 0.5 \right) \times \\ &\prod_{l=1}^L \text{Student's }t\left({\gamma_o}_l \mid 3, 0, 0.5 \right) \times \prod_{a=1}^2 \text{Student's }t\left({\psi_o}_a \mid 6, 0, 0.25 \right) \times \\ &\prod_{r=1}^R \text{Student's }t\left({{\text{z}_r}_o}_n \mid 6, 0, 0.25 \right) \times \text{Half-Student's }t^+\left({\sigma_r}_o \mid 6, 0, 0.25 \right) \\ \\ \boldsymbol{\theta}_{nt} &= \text{softmax}\left(\boldsymbol{\eta}_{nt}\right) = \frac{e^{\boldsymbol{\eta}_{nt}}}{\sum_{o=1}^O e^{{\eta_{nt}}_o}} \\ {\eta_{nt}}_o &= \begin{cases} 0, & o = 1, \\ \textbf{X}_{n\cdot} {\boldsymbol{\beta}}_o + \textbf{W}_{nt\cdot} {\boldsymbol{\gamma}}_o + A_o \sin \left(\frac{2\pi}{P_n} D_{nt} + \phi_o \right) + \sum_{r=1}^R {{\nu_r}_o}_n, & o \ne 1 \end{cases} \\ A_o &= \Vert \boldsymbol{\psi}_o \Vert \\ \phi_o &= \tan^{-1}\left( \frac{{\psi_o}_2}{{\psi_o}_1} \right) \\ {{\nu_r}_o}_n &= {\sigma_r}_o \times {{\text{z}_r}_o}_n \\ P_n &= \begin{cases} 365, & \text{nest }n\text{ not initiated in a leap year} \\ 366, & \text{nest }n\text{ initiated in a leap year} \end{cases} \end{align}\]} \end{align}