I'm reading documents about time series analysis, including Autoregressive–moving-average model, Moving average model, Wold's theorem, etc.
The notation MA(q) refers to the moving average model of order q:
$$X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q} \qquad (*)$$
where μ is the mean of the series, the $θ_1$, ..., $θ_q$ are the parameters of the model and the $ε_t$, $ε_{t−1}$,..., $ε_{t−q}$ are white noise error terms.
1) I would understand if we would want to express $X_t$ as a linear combination of past values $X_{t-1}$, $X_{t-2}$, $X_{t-3}$, etc. i.e. :
$$X_t = a_1 X_{t-1} + a_2 X_{t-2} + ... + a_q X_{t-q} + \epsilon_t$$
and if we would want to find optimal values for $(a_i)_{i \leq q}$.
This would make sense to me (by the way, does this approach exist, what's its name?).
2) But here I don't understand why we try to express $X_t$ in terms of past values of something which is totally uncontrolled and has nothing to do with $(X_t)$ : $\epsilon_t$, i.e. some random white noise!
Why do we do this in the moving average model? (By the way, how are exactly defined the $\epsilon_t$ ?)