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

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

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

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 8.2.3 for an example of this method.

Baggenstoss 2017-05-19