I have two physical models $f(\theta)$ and $g(\theta)$ (not probability distributions) parameterized on the same set of parameters $\theta$. I also have data $y$ with measurement noise $\epsilon$ which can be used to obtain posterior estimates $p(\theta | y)$ through one of the models:
\begin{align}
  y &= f(\theta) + \epsilon \\
  \text{where}\ \epsilon &\sim \text{Normal}(0, \sigma)
\end{align}
so a Gaussian maximum likelihood function can be defined:
\begin{equation}
p(y | \theta) \sim \exp\left(\frac{-(y - f(\theta))^2}{2\sigma^2} \right)
\end{equation}

Questions:

 1. Most of what I've read seems to be concerned with constructing posterior
    predictive distributions for new $y$ (denoting as $y_{\text{new}}$),
    as in $p(y_{\text{new}}|y) = \int p(y_{\text{new}}|\theta)
    p(\theta|y)d\theta$. But don't I just want the distribution $f(\theta)p(\theta|y)$
    without the measurement noise $\epsilon$?
 2. Letting $z=g(\theta)$, is it necessary to define a probability density $p(z|\theta)$ and define new predictions of $z$ as the marginal distribution $p(z|y) = \int p(z|\theta)
    p(\theta|y)d\theta$ (e.g., Robert, *The Bayesian Choice*, 2007)? Can I not just construct a probability distribution be constructed for the second
    physical model from $g(\theta)p(\theta|y)$?