If the question is whether there is a way to compute the expected loss $l$, or risk $R(\hat{f})=\mathbb{E}[l(Y, \hat{f}(X))]$ of an estimator $\hat{f}$ picked at random in some hypothesis space $\mathcal{H}$, then this cannot be done since the risk is an expectation over the true distribution of the data which one does not know.
Instead the empirical risk $\hat{R}(f) = \frac{1}{N} \sum_i l(y_i, f(x_i))$ is used as an approximation, i.e. one computes the error over the data. Minimisation of this risk is what most iterative algorithms do. PCA projections can be recast as empirical risk minimisation with squared loss under a linear model, I assume doing the same for PPCA is possible (see Bishop's paper on PPCA). There are many bounds on how well the empirical risk approximates the true risk (so-called generalisation bounds $|R(\hat{f})-\hat{R}(\hat{f})|$).
If the question is whether there are a priori bounds to estimate $\mathbb{E}[\hat{R}(\hat{f})]$ when picking $\hat{f} \in \mathcal{H}$ at random, where this expectation would not be wrt. datasets as is usually the case for the expected empirical risk of an estimator, but wrt. the distribution selecting $\hat{f}$, then I guess some computation is possible for each specific case. E.g. if we take squared loss and $\mathcal{H}$ to be parameterized by $\theta$, and we pick $\theta \sim \Theta$ then by linearity one has to compute
$$\frac{1}{N}\sum_i \mathbb{E}_{\theta \sim \Theta}[(y_i - \hat{f}_{\theta}(x_i))^2]$$
And this might be possible or not depending on the choice of $\mathcal{H}$.
