Let there be a random variable with the following properties
$Y_{t} = \mu + \beta Y_{t-1} + \epsilon_{t} \quad\text{such that} \quad \epsilon_{t}\sim \mathcal{N}(0,\sigma^{2})$
Estimate the parameters by the maximum likelihood method.
My doubt per se isn't about the inference part, I'm interested in one of the steps in-between. When computing the likelihood of a random sample $\{ X_{t} \}_{t=1}^{N}$, we can expand the joint into a product of conditional densities as in
$f_{X_{1},...,X_{N}}=f_{X_{1}}f_{X_{2}|X_{1}}f_{X_{3}|X_{2},X_{1}}...f_{X_{N}|X_{N-1},...,X_{1}}$
and we know by definition that
$f_{X|Y}=\frac{f_{X,Y}}{f_{Y}}$
However I see in some examples on the web that people simply take $Y_{t-1}=y_{t-1}$ as fixed and substitute in the equation to get the new mean $\mathbb{E}(Y_{t}|Y_{t-1}=y_{t-1})= \mu + \beta y_{t-1}$, instead of the regular $\frac{\mu}{1-\beta}$.
My question is a bit naive. When can we simply substitute like that (Is it correct?) and when do we calculate the conditional density by definition?