An intuitive motivation for the Innovations Algorithm used in Time Series Is there a intuitive motivation for the Innovations Algorithm used in Time Series for forecasting and parameter estimations? I understand that one objective of the algorithm is to be able to write the linear predictors with uncorrelated terms (the innovations). But why would that be desirable, and how would that translate in the actual algorithm that we see in books like «Introduction to Times Series and Forecasting» by Brockwell and Davis?
Any help would be appreciated.
Addendum: To avoid having this question closed for «being unclear what I'm asking» , and after some searching, I've found these slides. So, from slides 4 to 7 there is equation (1) with several equalities, one of which I don't understand. The first equality is from uncorrelated property of the innovations. However, where does the second equality come from?
 A: I'd say the biggest advantage of using the innovations algorithm is that it's much easier to invert a huge diagonal covariance matrix instead of a huge nondiagonal covariance matrix. Say you have some data $\mathbf{X} = (X_1,\ldots X_n)'$. Evaluating a Gaussian likelihood directly involves inverting a really big covariance matrix which might be a nasty function of all your model parameters. The innovations $\mathbf{U} = (U_1,\ldots,U_n)'$ (where $U_i = X_i - \hat{X}_i$) have a diagonal covariance matrix. This is problem 2.20 in the book. You can prove it using the linear relationship between these two vectors along with some related properties, or you can use properties of innovations and projections. The quadratic form will turn into a single sum, and the determinant will be a product of $n$ numbers, it's great.
In the case of forecasting, if you're using it for a pure MA process, the amount of coefficients $\{\theta_{nj}\}_{j=1}^n$ that you weight all the past innovations by, is not increasing in time ($n$). This is because you get more and more $0$s. For an MA(q) process, you only need the coefficients $\theta_{n1},\ldots, \theta_{nq}$, which are the same as the model parameters $\theta_1, \ldots,\theta_q$. This is what the authors' notation was suggesting: that if you use an innovations algorithm with a pure MA process, the coefficients will bear a striking resemblance to the model parameters (it's the same situation with the pure AR models and the Durbin-Levinson algorithm).
Edit:

However, where does the second equality come from?

The second equality also comes from the uncorrelated properties of innovations.
Because 
\begin{align*}
<\hat{X}_{n+1}, X_{n+1-j} - \hat{X}_{n+1-j}> &= <\sum_{i=1}^n\theta_{ni}(X_{n+1-i} - \hat{X}_{n+1-i}), X_{n+1-j} - \hat{X}_{n+1-j}> \\
&=\sum_{i=1}^n\theta_{ni} <X_{n+1-i} - \hat{X}_{n+1-i}, X_{n+1-j} - \hat{X}_{n+1-j}> \\
&=\theta_{nj} <X_{n+1-j} - \hat{X}_{n+1-j}, X_{n+1-j} - \hat{X}_{n+1-j}>.
\end{align*}
