I was reading the Meta Temporal Point Processes paper and was having trouble understading the training objective presented. The authors state that it is the ELBO used in Variational Inference techniques.

The ELBO training objective

where $L$ is the number of observations, $\tau_{l}$ is a continuous varaible observed $l$th, and $z \sim p_{\theta}\left(z | \mathcal{C}_{L}\right)$ according to the paper, where $C_{l}$ is defined as $\left\{ \tau_{t-k+1:t} \right\}_{t=1}^{l-1}$.

However, the above expression is very confusing, and I have three questions.

  1. $p_{\theta}$ is defined as the distribution of $z$, however, it is used as the distribution for $\tau$ as well?
  2. Why is $p_{\theta}\left(\tau_{l}\right)$ not $0$ given that $\tau$ is a continuous variable?
  3. Why did $\tau_{l+1}$ in the first term become $\tau_{l}$ from $(6)$ to $(7)$?


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