Data model

Suppose the data is generated from a model ${\bf x}={\bf H}_\theta {\bf a}+ {\bf v},$ where ${\bf H}_\theta$ is a $N\times P$ matrix of basis functions, ${\bf a}$ is a $P\times 1$ amplitude vector, and ${\bf v}$ is an $N\times 1$ vector of iid zero-mean Gaussian RVs with variance $\sigma^2$. Assume each column of ${\bf H}_\theta$ depends non-linearly on a parameter $\theta_i$. Let $\theta$ be these parameters. This model underlies many important estimation problems, such as the estimation of the frequency of sinewaves in noise [30].

The likelihood function is given by

$$\log p({\bf x};$$   $\theta$$$,{\bf a},\sigma^2)= -\frac{N}{2}\log(2\pi\sigma^2)- \frac{1}{2\sigma^2}({\bf x}-{\bf H}_\theta {\bf a})^\prime ({\bf x}-{\bf H}_\theta {\bf a}).$$