## Appendix C Learning Parameters  
We learn parameters given the training data using the gradient descent method. Note that for maximum likelihood estimation (ML), we treat the field affinity as another parameter (i.e., APP without a prior), while for maximum a posteriori (MAP) we impose a prior distribution and optimize its hyperparameters instead (i.e., APP with a prior).  
**APP without a prior.** Without a prior distribution, the objective function to maximize is the log likelihood distribution of all the citation timestamps \(\mathbf{D}\) of the cited field,  
\[L_{\text{ML}}=\ln p(\mathbf{D}\mid\xi)=\sum_{f\in F}\ln p\left(\mathbf{D}^{f}\Bigm{|} \xi^{f}\right)\ . \tag{117}\]  
Then, the partial derivatives for each parameter are  
\[\frac{\partial L_{\text{ML}}}{\partial\xi^{f}} =\sum_{i=1}^{I}\left(\frac{N_{i}^{f}}{\xi^{f}}-M_{i}^{f}\right)\ \, \tag{118}\] \[\frac{\partial L_{\text{ML}}}{\partial\mu_{i}^{f}} =\frac{1}{\sigma_{i}^{f}}\left(\xi^{f}\cdot\left(n_{0}+N_{i}^{f} \right)\cdot\phi\left(\tau_{i}^{f}\right)+\sum_{n=1}^{N_{i}^{f}}\left(\tau_{n }^{f}-\xi^{f}\cdot\phi\left(\tau_{n}^{f}\right)\right)\right)\ \,\] (119) \[\frac{\partial L_{\text{ML}}}{\partial\sigma_{i}^{f}} =\frac{1}{\sigma_{i}^{f}}\left(\xi^{f}\cdot\left(n_{0}+N_{i}^{f} \right)\cdot\tau_{i}^{f}\cdot\phi\left(\tau_{i}^{f}\right)+\sum_{n=1}^{N_{i}^ {f}}\left(\left(\tau_{n}^{f}\right)^{2}-\xi^{f}\cdot\tau_{n}^{f}\cdot\phi\left( \tau_{n}^{f}\right)\right)-N_{i}^{f}\right)\ . \tag{120}\]  
**APP with a prior.** In Bayesian estimation, we usually learn hyperparameters by maximizing the log marginal likelihood as  
\[L_{\text{MAP}}=\ln p(\mathbf{D})=\sum_{f\in F}\ln\int p\left(\mathbf{D}^{f}\Bigm{|}\xi ^{f}\right)p\left(\xi^{f}\right)d\xi^{f}\ . \tag{121}\]  
Substituting Equations (106) and (113) into Equation (121) and taking partial derivatives with respect to hyperparameters gives  
\[\frac{\partial L_{\text{MAP}}}{\partial\alpha} =\sum_{f\in F}\left(\psi_{0}(\alpha^{\prime})-\psi_{0}(\alpha)+ \ln\beta-\ln\beta^{\prime}\right)\ \, \tag{122}\] \[\frac{\partial L_{\text{MAP}}}{\partial\beta} =\sum_{f\in F}\left(\frac{\alpha^{\prime}}{\beta^{\prime}}-\frac {\alpha}{\beta}\right)\ \, \tag{123}\]  
where \(\psi_{0}(\cdot)\) denotes the digamma function, the logarithmic derivative of the gamma function.  
Similarly, we take the partial derivatives with respect to \(\theta_{i}^{f}\) as  
\[\frac{\partial L_{\text{MAP}}}{\partial\mu_{i}^{f}} =\frac{1}{\sigma_{i}^{f}}\Bigg{(}\frac{\alpha^{\prime}}{\beta^{ \prime}}\left(n_{0}+N_{i}^{f}\right)\phi\left(\tau_{i}^{f}\right)+\sum_{n=1}^ {N_{i}^{f}}\left(\tau_{n}^{f}-\frac{\alpha^{\prime}}{\beta^{\prime}}\phi\left( \tau_{n}^{f}\right)\right)\Biggm{)}\ \, \tag{124}\] \[\frac{\partial L_{\text{MAP}}}{\partial\sigma_{i}^{f}} =\frac{1}{\sigma_{i}^{f}}\Bigg{(}\frac{\alpha^{\prime}}{\beta^{ \prime}}\left(n_{0}+N_{i}^{f}\right)\tau_{i}^{f}\phi\left(\tau_{i}^{f}\right) +\sum_{n=1}^{N_{i}^{f}}\left(\left(\tau_{n}^{f}\right)^{2}-\frac{\alpha^{ \prime}}{\beta^{\prime}}\tau_{n}^{f}\phi\left(\tau_{n}^{f}\right)\right)-N_{i }^{f}\Biggm{)}\ \, \tag{125}\]  
where \(\tau_{i}^{f}=\left(\ln T-\mu_{i}^{f}\right)/\sigma_{i}^{f}\), \(\tau_{n}^{f}=\left(\ln t_{n}^{f}-\mu_{i}^{f}\right)/\sigma_{i}^{f}\), and \(\phi(\cdot)\) is the probability density function of the standard normal distribution. Note that \(n_{0}\) is the prior citations for each timeline. Refer to Appendix D for details.