# Partial Parameter Estimation (MLE/LMMSE)

I have a basic question. MLE/LMSSE is introduced as follows: $$Y = H\theta + W$$ where $H$ is the linear model matrix, $W$ is measurement noise (let's assume it is normal so MLE = LMSSE). $\theta$ is the vector of parameters.

The ML estimate is well-known: $(H'H)^{-1}H'\bf{Y} = \widehat{\theta}\quad\quad$ ($\bf{Y}$ is the vector of observations.)

Let's say I am only interested estimating one of the parameters (say the first parameter).
Is it equal to the first entry of $\widehat{\theta}$?

More generally is the ML estimate of $S\theta$ where $S$ is some low-rank matrix the same as $S\widehat{\theta}$? I do not think that is the case, but I don't know how exactly to deal with it.

• Cross posted here: math.stackexchange.com/questions/858695/… – Kirill Jul 7 '14 at 7:29
• I suggested to close the cross-posted question above on math (first voter). We should try to not close it on both sites. – gnometorule Jul 7 '14 at 7:52

The posterior distribution over $\theta$ is Gaussian here.
When you apply the linear function $\theta\mapsto S\theta$ to it, the new random variable $S\theta$ is still Gaussian, so its MLE is just its expectation, and $\mathbb{E}[S\theta] = S\mathbb{E}[\theta]$.
• @user161378 The way you phrased the question is that you estimate the whole model (all $\theta$'s), then take a subset, which is what I answered. If instead you change your model to only consider one of the parameters, then of course the estimate will be different; the new model matrix will be just the first column of the original matrix. – Kirill Jul 7 '14 at 19:36