Conditional probability of timeseries given history

Question

How is the following interpreted?

Given two time series $X = \{x_1, \dots, x_n\}$ and $Y=\{y_1, y_2, \dots, y_n \}$, calculate probability of the next $x$, given the history $\mathbf y_d = \{ y_{n-d+1}, \dots, y_{n-1}, y_n \}$ of $y$:

$$ P(x_{n+1} | \mathbf{y_d}) = \frac{P(x_{n+1},\mathbf y_d)}{P(\mathbf y_d)} $$ Is this the same as calculating $$\frac{P(x_{n+1}, y_{n-d+1}, \dots, y_n)}{P(y_{n-d+1}, \dots, y_n)}$$ or is there some "trick" in estimating the probability of finding the specific sequence $\mathbf y_d$ somewhere in $Y$?

Answer 1

What you write is simply the formula for conditional probability. And, there is no general trick. There may be special cases though. Simplest of them is first order markov process, with output equal to state, i.e. $x_n=y_n$. Then, $$p(x_{n+1}|\mathbf{y_d})=p(x_{n+1}|y_n)$$ So, you can have tricks depending on the dependence graph.