Why is ancestral sampling used in autoregressive models? I have been reading about autoregressive models. Based on what I've read, it seems to me that all autoregressive models use ancestral sampling. For instance, this paper says the following in Abstract:

We introduce a deep, generative autoencoder capable of learning hierarchies of distributed representations from data. Successive deep stochastic hidden layers are equipped with autoregressive connections, which enable the model to be sampled from quickly and exactly via ancestral sampling.

However, what I don't understand is why (as I understand it) all autoregressive models use ancestral sampling. Why is ancestral sampling used in autoregressive models?

You can use ancestral sampling in all autoregressive models where the conditional probability distribution is known given the previous samples, e.g. $p(x_t \mid x_{t - 1}, \dots, x_1)$.
For example, consider a first-order Markov process:
$$x_1\rightarrow x_2\rightarrow \dots \rightarrow x_{t-1}\rightarrow x_t$$
The joint probability distribution would be $$p(x_1, \dots, x_t)=p(x_1)p(x_2 \mid x_1) \dots p(x_t \mid x_{t - 1})$$
Assume you know all the conditional probabilities/densities here. Then, you can generate an autoregressive process via first sampling $x_1$, then sampling $x_2$ from $p(x_2 \mid x_1)$, then $x_3$ from $p(x_3 \mid x_2)$, and so on, reaching up to $x_t$. You start sampling at the very ancestor and keep moving up.
