Can't think of a more accurate title, so I'll illustrate the problem with an example.
I want to record temperature using cheap noisy sensors. I also have recordings from a gold-standard reference point, so I can model the gold-standard temperature and use it in the future to estimate the temperature more accurately than just using my sensors' outputs directly.
For each reference recording $y_i$, I have my cheap sensor recordings $x_{ij}$ where $j = 1,...5$, so I can make a varying slope & intercept hierarchical model (plus group priors on $\alpha$ and $\beta$). Would this be a valid approach? In most examples I've seen both the outcome and predictor vary for each observation, whereas in this model $y_i$ is repeated for 5 levels of $j$.
$$y_i \sim N(\mu_{ij}, \sigma_{i})$$ $$\mu_{ij} = \alpha_j + \beta_j x_{ij}$$
When using this live I could get 5 posteriors of $y$ for each timepoint which can be combined to generate my temperature estimate.
Say I notice that there is interference with relative humidity on my temperature sensors which I need to adjust for. I then buy 3 relative humidity sensors and I save that data in $z_{ik}$ where $k = 1, ..., 3$.
If I took the same approach as before and modelled each permutation of $i$, $j$ and $k$, I'd end up with using each $y_i$ 15 times now, which again just doesn't seem right.
$$y_i \sim N(\mu_{ijk}, \sigma_{i})$$ $$\mu_{ij} = \alpha_j + \beta_j x_{ij} + \gamma_k z_{ik}$$
