Autoregressive models

Model the probability of subsequent elements using previous observations or graphically, can be interpreted as fully connected DAG. By chain rule of probability:

$$\begin{array}{c}p(x_{1:T})=p(x_1)p(x_2|x_1)\cdots =\prod _{t=1}^{T}p(x_t|x_{1:t-1})\end{array}$$

Above equation can be relaxed in various ways to treat the intractability of conditional dependence:

• Markov assumption: $p(x_t|x_{1:t-1})=p(x_t|x_{t-1})$
• N-gram model: $p(x_t|x_{1:t-1})=p(x_t:x_{t-n-1:t-1})$
• Hidden state $z_t$: compress past into hidden state
  • when $z_t$ is a deterministic function of past states, resulting model is RNN
  • when $z_t$ is stochastic function, resulting model is hidden markov model.