Model Comparison: How to constrain a model to positive mu-differences?

[junpenglao]

@Olaf

It’s a motoric task measuring 2 end effector positions, thus forming a trial-to-trial point cloud in a cartesian 2D-coordinate system. The demeaned 2D-coordinate vectors get projected onto a vector and its orthogonal, basically just a change of basis.

Since they are orthogonal you can model them with univariate distribution that has similar shape and the same support (e.g., positive only) as your observed.

@Olaf

By flattening you mean throwing all participants together? I don’t quite get your code example. mu and sigma are of shape n_users, but you only estimate these for 1 user and using the data of all users at the same time?

Usually, you can coded your data as a table like:

subject_id | condition1 | condition2 | observation
         0 |   parallel |    block_1 |       .3452
...

Instead of having 1 column per participant and trials being the rows in the observed data, you have 1 column for all your data, and participant and trials are indicator (subject_id and trial_id).

Then in your model, you can either use GLM like approach to construct a design matrix, or use indexing:

subject_mu = pm.Normal(..., shape=n_subject)  <= shape is n_subject
subject_mu_all_trial = subject_mu[subject_id] <= shape is now number of row in your data frame

which estimate the parameter independently for each subject.

No matter what approach you decide to go with in terms of model comparison / hypothesis testing, I think you should use this formulation as it is a more general way to do so, and connect nicely with classical GLM formulation. You can also consult the recent book Regression and other stories