# How to model repeated measurements with the same outcome in a Bayesian framework?

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.

$$y_i \sim N(\mu_{ijk}, \sigma_{i})$$ $$\mu_{ij} = \alpha_j + \beta_j x_{ij} + \gamma_k z_{ik}$$

• Can you please give a more detailed description of the system you want to model? Are you modelling the recordings of 5 sensors? Or 5 time-points for one cheap sensor? – user289381 Jul 12 '20 at 21:36
• Ok, so if I have understood correctly, you want to model the gold standard using the cheap sensors, ostensibly so that you can just use the data you collect to predict the gold standard. Is that correct? – Demetri Pananos Jul 12 '20 at 23:05
• @DemetriPananos is right. So I have a training set where for each time-point I have 6 recordings, 1 for each of the 5 cheap sensors + 1 from the gold-standard. Then in the future I won't have access to the gold-standard, just the 5 cheap sensors. – Stuart Lacy Jul 14 '20 at 16:03