- We model the log-hazard as \(\log h(t|x) = \beta_0(t) + \sum_{j = 1}^{p}x_j \beta_j(t) \varphi(\sum_{i\neq j} w_{ij} x_i)\)
- We choose the 'activation function' \(\varphi(\cdot)\) to be a logistic sigmoid in keeping with standard NN practice; i.e. \(\varphi(\cdot) = \frac{2}{1+\exp(-x)} = 2\textrm{logit}^{-1}(x)\).
- We have additivity whenever the \(w_{ij} = 0\) and proportionality whenever the \(\beta_j\) are constant.
- This suggests a model where the \(w_{ij}\) are given LN-CASS priors, and the \(\beta_j(t)\) are modelled as basis function expansions with hierarchical LN-CASS priors on the coefficients.
- \(\beta_0(t)\) should probably also be modelled nonparametrically, although its complexity is not of great concern.
- The likelihood is then \(\prod_{i=1}^nh(t_i)^{\delta_i}S(t_i) = \prod_{i=1}^nh(t_i)^{\delta_i}\exp(-H(t_i))\).
- So in order to compute the likelihood, we need to compute the survival function, or alternatively the cumulative hazard.
- Writing out the hazard, the integral we need to compute is:
\(H(t_i) = \int_{0}^{t_i} \exp\left( \beta_0(\tau))\exp( \sum_{j = 1}^{p} \varphi_j x_j \sum_{k = 1}^{m}\omega_{jk}\psi_k(\tau) \right) d\tau\)
- Ideally we would like this integral to have a closed form representation.
- Alternatively, working with the log-likelihood:
- \(\ell(\theta) = \sum_{i = 1}^{n} \log(h(t_i)^{\delta_i}) + \log(S(t_i)) = \sum_{i = 1}^{n} \log(h(t_i)^{\delta_i}) - H(t_i)\)
- The baseline hazard can be flexibly modelled using the low-rank thin plate splines of Murray et al. (2016, Bayes. Anal.)
- Given a discretisation of the time domain \(t_1,t_2,...\),