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 $Y$?