\begin{table*}[!t] 
    \vspace{-0.0in}
    \setlength\tabcolsep{0.8pt}
    \scriptsize
    \caption{Robustness Transformations: Original and Transformed Models ($\alpha, \beta, s, \eta, \nu, \rho, \sigma, z$ are parameter variable placeholders; $y$ is a data variable placeholder; $F, \pi$ are distribution placeholders)}
    \label{tab:transformations}
    \vspace{-0.03in}
    \begin{tabular}{|c|c|c|c|c|}\toprule       
    \begin{minipage}{0.18\textwidth}
        \scriptsize
        \centering
        \begin{tabular}{c}
          $\beta \sim \pi_{\beta}(\alpha)$ \\
          $y_{i=1}^{D}\sim F(\beta)$ \\[.05in]
          $\Downarrow$ \\[.05in]
          $w_{i=1}^{D} \sim \textit{Beta}(\gamma, \zeta)$ \\
          $\beta \sim \pi_{\beta}(\alpha)$\\
          $y_{i=1}^{D}\sim F(\beta)^{w_i}$ \\
      \end{tabular}
      \end{minipage} 
      &
      \begin{minipage}{0.16\textwidth}      
      \scriptsize\centering
      \begin{tabular}{c}  
       $\beta \sim \pi_{\beta}(\alpha)$ \\ 
       $y_{i=1}^{D}\sim F(\beta)$   \\[.05in]
       $\Downarrow$ \\[.05in]
       $\beta \sim \pi_{\beta}(\alpha)$ \\   
       $s \sim \textit{Unif}(0, 1)$\\
      $\eta_{i=1}^{D} \sim \mathcal{N}(\beta, s)$  \\      
      $y_{i=1}^{D}\sim F(\eta_i)$  \\    
        \end{tabular}      
        \end{minipage}
    &
        \begin{minipage}{0.2\textwidth}
          \begin{center}
            \scriptsize
            \begin{tabular}{c}
              $\beta \sim \pi_{\beta}(\alpha)$\\
              $y_{i=1}^{D}\sim\mathcal{N}(\beta, \sigma)$  \\[.05in]
              $\Downarrow$\\[.05in]
              $\nu \sim \pi_{\nu}(\gamma)$ \\
              $\beta \sim \pi_{\beta}(\alpha)$\\
              $y_{i=1}^{D}\sim\mathcal{T}(\nu,\beta, \sigma)$\\
          \end{tabular}
          \end{center}  
          \end{minipage}  
        &
          \begin{minipage}{0.2\textwidth}
            \begin{center}
              \scriptsize
              \begin{tabular}{c}
                $\beta \sim \pi_{\beta}(\alpha)$ \\
                $y_{i=1}^{D} \sim \mathcal{N}(\beta, \sigma)$ \\[.05in]
                $\Downarrow$\\[.05in]
               $\nu \sim \pi_{\nu}(\gamma)$\\
               $\tau_{i=1}^{D} \sim \textit{Gamma}(\frac{\nu}{2},\frac{\nu}{2})$\\
             $\beta \sim \pi_{\beta}(\alpha) $ \\
             $y_{i=1}^{D}\sim \mathcal{N}(\beta, \frac{\sigma}{\sqrt{\tau_i}}\cdot)$\\
              \end{tabular}
            \end{center}
            \end{minipage}
            &
            \begin{minipage}{0.22\textwidth}  
              \scriptsize
              \begin{center}
              \begin{tabular}{c}
                $\beta \sim \pi_{\beta}(\alpha)$\\
                $y_{i=1}^{D}\sim F(\beta,\sigma)$\\[.05in]
                $\Downarrow$\\[.05in]
              $\rho_{\textit{out}},\eta_{\textit{out}},\sigma_{\textit{out}}\sim\pi(\gamma)$\\
             $\nu\sim\mathcal{N}(\eta_{\textit{out}},\sigma_{\textit{out}})$\\
             $\beta \sim \pi_{\beta}(\alpha) $ \\
            $z_{i=1}^{D} \sim \textit{Bernoulli}(\rho_{\textit{out}})$\\
              $y_{i}|z_i=0\sim F(\beta,\sigma)$\\
              $y_{i}|z_i=1\sim F(\beta,\sqrt{e^{\nu}})$\\
              \end{tabular}
              \end{center}
              \end{minipage} \\
              \midrule
              (a) Reweighting&(b) Localization&(c) Normal-to-Student-T& (d) Reparameterization&(e) Cont. Mixture\\\bottomrule
            \end{tabular}
\vspace{-.00in}           
  \end{table*}
