1.4 Objective Function

The goal is to adjust $\mathbf{x}$ so that the likelihood of the observations is maximized:

$$p\left(\mathbf{z}\vert\mathbf{x}\right)\propto\exp\left(-\frac{1}{2}\mathbf{v}^{T}\cdot\mathbf{R}^{-1}\cdot\mathbf{v}\right)$$ (18)

As the logarithm is monotonic, this is equivalent to minimizing the negative log-likelihood objective function:

$$L=\mathbf{v}^{T}\cdot\mathbf{R}^{-1}\cdot\mathbf{v}$$ (19)

When the individual observations are independent, the covariance matrix $\mathbf{R}$ is block diagonal. Then the objective function reduces to a sum over the observations (again as a function of $\mathbf{x}$):

$$L_{i} \equiv \mathbf{v}_{i}^{T}\cdot\mathbf{R}_{i}^{-1}\cdot\mathbf{v}_{i}$$ (20)

$$L = \sum_{i}L_{i}$$ (21)

The value of this objective function for some specific parameters $\mathbf{x}$ is often called the residual.