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 [24].

The likelihood function is given by

$\displaystyle \log p({\bf x};$   $ \theta$$\displaystyle ,{\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}).$



Baggenstoss 2017-05-19