UMS

We now consider UMS - that is to generate samples of data which will have exactly the specified ML parameter estimates, uniformly on the manifold. Given fixed MLE values of $\hat{\mbox{\boldmath $\theta$}}$, $\hat{{\bf a}}, \; \hat{\sigma^2}$, the we can easily define the manifold of all ${\bf x}$ that lead to the given MLE as follows. Define the statistics ${\bf z}_0={\bf x}^\prime {\bf x}$, $\;\;{\bf z}_1={\bf H}_\theta^\prime {\bf x}$, and ${\bf z}_2=\hat{{\bf a}}^\prime {\bf H}_\theta^{\theta \prime} {\bf x}$. We can use (4.2), and (4.3) to compute ${\bf z}_0,{\bf z}_1$. The ML estimates $\hat{\mbox{\boldmath $\theta$}},\hat{{\bf a}},\hat{\sigma^2}$ are such that the partial derivative of $\log p({\bf x};$   $\theta$$,{\bf a},\sigma^2)$ with respect to each parameter is zero. The derivative constraint for $\hat{\mbox{\boldmath $\theta$}}$ leads to

$${\bf x}^\prime {\bf H}_\theta^{\theta}{\bf a}= {\bf a}^\prime {\bf H}_\theta^\prime {\bf H}_\theta^{\theta}{\bf a}.$$ (4.7)

We can use (4.7) to compute ${\bf z}_2$. So, we are able to compute ${\bf z}_0,{\bf z}_1,{\bf z}_2$ just from $\hat{\mbox{\boldmath $\theta$}},\hat{{\bf a}},\hat{\sigma^2}$. The equations defining ${\bf z}_0,{\bf z}_1,{\bf z}_2$ lead to a set of constraints for ${\bf x}$ that can be written in the form

$${\bf A}^\prime {\bf x}=[{\bf z}_1 \; {\bf z}_2]^\prime, \;\;\;{\bf x}^\prime{\bf x}={\bf z}_0$$

where ${\bf A}=[{\bf H}_\theta \;\; \hat{{\bf a}}^\prime {\bf H}_\theta^{\theta}].$ This is the problem of Section 4.4. We can apply the results of that Section to sample the manifold. Every sample will meet the derivative constraint for $\theta$ as well as produce the same amplitude and variance estimates, so will produce the given ML solution. See software/test_ml.m for an example of sinusoidal frequency estimation. See also Section 9.2.3 for an example of this method.