I want to calibrate the parameters $\theta$ of a known forward model $y=f(\theta, x)$, i.e., I want to identify offsets of around +- 10% from a nominal model $\theta_0$
I can not measure $y$ directly, but my measurement function $h_c$ only depends on this forward model: $y_c = h_c(f(\theta, x))$ Let's call the calibration set $C$, consisting of $N_C$ pairs $(y_c, x_c)$.
When I identify the $\theta$ via a MAP approach or some variant of Least Squares, I minimize the error in $h_c$ over $C$. But I want to achieve something else.
Instead, I want to minimize the error of a test function $y_t = h_t(\theta, x)$ over a set $T$ of pairs $(y_t, x_t)$. However, my measurement setup does not allow me to collect measurements for this directly.
To summarize, I have to measure functions. For $h_c$, I can collect measurements $C$ and perform the calibration. But I care about minimizing the error of $h_t$ over $T$. All the functions and sets are known. However, they do not model the real system perfectly and there are epistemic errors.
Are there any strategies for this kind of scenario? I am also looking for literature on this topic.