add plot_lm

[OriolAbril]

Add a simplified version of plot_lm in current ArviZ. I think to start we could limit ourselves to something with 3 elements only: CI band, mean line and observation scatter. For example the plot from this section https://bambinos.github.io/bambi/notebooks/distributional_models.html#constant-alpha (adding link in addition to copy of the image so you have code to generate data like that to test the function, it might not match all the assumptions below though but it has the data and reorganizing it should be easier than generating it I think):

We need to access potentially 4 different data elements:

• observations for target variable. It should be in idata.observed_data
• observations for independent variable. It should be in idata.constant_data in general, but I think we could try allowing string/list (access from constant data) or dataarray/dataset to use directly.
• predictions/posterior predictive. I think as default we can get the same variables there are in observed_data group from the posterior_predictive group. I would restrict this to string/list only. But we might want to allow posterior_predictive/y/predictions/y to allow indicating which groups holds the data we want to plot. We'd then compute the point estimate and credible interval according to rcParams
  • open question: cc @aloctavodia do we want to integrate smoothing here? or support smoothing only if done manually and then pre-computed summaries are provided via stats dict?
• independent variable data for ci/mean. I think this should be optional and default to the observations for independent variable, but it should be possible to change. One automatic change in default could be when using data from predictions group, get data with same same as independent variable observations but from predictions_constant_data instead. And like the observation analog, it should also allow dataarray/dataset.

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I would start allowing to generate only one plot in the figure. After that we can discuss how to incorporate faceting/mappings to that for multiple outcomes, multiple covariates, both at the same time?

[The-Broken-Keyboard]

assign it to me

[aloctavodia]

Yes, I think we should integrate smoothing here. If plot_lm turns out to be useful enough, there will be little reason to "manually" compute a smoothed hdi, unless someone wants something very customized. Besides smoothing, when plotting a regression as a line or band, we need the x-values to be ordered. It would be nice to have a function like plot_lm that handles all that with a single call.

I agree it's a good idea to start simple. So I agree with what you have already mentioned @OriolAbril. One feature that we would like to have is to be able to plot more than one ci_band. This could be more than one band for the same data (e.g. 0.5 and 0.9 hdi for the predicted data). But also for different data (e.g. 0.9 for regression mean (the "mu" parameter), and 0.9 for predicted data).

Not sure how flexible we want plot_lm to be. But it could help to check existing plots in related libraries, for instance, in PyMC-BART we have plot_pdp and plot_ice (which plots overlayed single predictions). Bambi can also generate a plot in the same spirit. It would be nice if pymc-bart restricts itself to computing the partial dependence and then feeds the result to plot_lm, but of course, we can rewrite the plots in this library to internally use arviz-plots even without plot_lm, it's just a matter of which level of abstraction to use.