What to call this intermediate step in my maximum a posteriori calculation? I have calculated a posterior
$p(P_1, P_2 | D)$
where $P_1$ has a few and $P_2$ has many dimensions. In the process of calculating the maximum a posteriori estimate for $(P_1,P_2)$ given certain data $D$, I calculate the maximum projection along all dimensions of $P_2$, i.e. 
$f(P_1,D) = \underset{P_2}{max} [p(P_1, P_2 | D)]$
which, given my data $D$, is a function of $P_1$ that I can numerically maximize to find the $P_1$ where the posterior reaches its maximum.
Is this something that is often done? Does $f$ have a name?
Edit: Even though I have accepted one answer, Simon in his answer makes a valid point that maximum a posteriori (MAP) in general is a problem, so don't stop reading early ;-)!
 A: There is a slight problem with this construction: it doesn't exist. Well, it's not uniquely defined without a further qualification.
The problem stems from the fact that, unlike maximum likelihood estimators, MAP estimators are not invariant under reparameterisation. That is, if we find a MAP estimate $\tilde\theta$ for some parameter $\theta$, and then reparamterise the problem to a different parameter $\psi = \psi(\theta)$, then the MAP estimate for $\psi$ may not be $\psi(\tilde\theta)$.
So, how does this relevant to your problem?
Suppose that your posterior for $(P_1,P_2)$ ends up being a bivariate normal with mean $(0,0)$ and covariance matrix $\begin{pmatrix}1 & \rho \\ \rho & 1 \end{pmatrix}$. Now, we have:
$$\max_{P_2} \ p(P_1,P_2|D) = p(P_1|D) \max_{P_2} p(P_2|P_1,D)$$
So we just need to find the mode of $P_2 | P_1,D$, which is a normal distribution $N(\rho P_1,1-\rho^2)$, whose mode is the mean $P_2 = \rho P_1$, so your function can be found by simple substitution, giving 
$$\max_{P_2} \ p(P_1,P_2|D) = \phi(P_1) p(P_2 =\rho P_1 , P_1|D) = \phi(P_1) \frac{1}{\sqrt{(2 \pi (1-\rho^2)}}$$
where $\phi$ is the density of the standard normal.
Okay, now suppose that instead we reparameterise our posterior so that we use $Q_2 = \exp(P_2)$. $Q_2 | P_1, D$ is a lognormal distribution, which has mode $\exp(\rho X_1 - (1-\rho^2))$. So substituting this in:
$$\max_{Q_2} \ p(P_1,Q_2 |D) = \phi(P_1) p(Q_2=\exp(\rho P_1 - (1-\rho^2)) \mid P_1,D) = \phi(P_1) \frac{\exp[(1-\rho^2)/2 - \rho P_1]}{\sqrt{(2 \pi (1-\rho^2)}} $$
which is a different function.
So the lesson is: you have to clarify the variable that you have profiled out.
A: Using the terminology of Profile Likelihood from the maximum likelihood estimation theory, your way of finding the $f$ may be called profile posterior distribution. 
A: I don't know of an official name, but I would call it something like 'the MAP conditional on P1'.
