Chapter 15 Likelihood Components with Warping:

Gaussian Process Likelihood: (Remains the same)

$$p(\eta_{t,k} | μ(t), k(t, t')) = \mathcal{N}(μ(t), k(t, t'))$$ Topic Distribution Prior: (Remains the same)

$$p(\theta_{d,t=0} | \alpha) = \text{Dirichlet}(\alpha)$$ Topic Distribution Update Likelihood (Incorporating Warping):

Conceptual: The likelihood of observing the new diagnoses given the topic distribution is now conditioned on the individual’s warped time. Representation: $$p(\theta_{d,t} | η_t, w_{d,t}, \gamma_d, ρ_d ) \propto \prod_{k=1}^K \left[\theta_{d, t-1, k} \cdot \prod_{v \in w_{d,t}} \pi_{v,k,W(t, ρ_d)} \right]^{\gamma_d}$$

Note the $W(t, \rho_d)$ in the subscript of $\pi_{v,k,W(t, \rho_d)}$, highlighting that disease probabilities are obtained from the warped timeline. Predictive Likelihood (Incorporating Warping):

Similarly, prediction depends on the warped time for the individual: p(w_{d,t+t} | θ_{d,t}, η_{t+t}, ρ_d) = {v w{d,t+t}} k {d,t,k} * _{v,k,W(t + t, ρ_d)} Key Points

Indirect Representation: The effect of warping is implicit in the modified likelihoods through the warped time index W(t, ρ_d). Procedural Implementation: Computationally, you’d likely have procedural steps to determine the appropriate values of η and π corresponding to the warped time of an individual.