I am reading the funsors paper and got stuck in a couple of questions.

In Figure 1, particularly the Gaussian defining p_x|y,z. I don’t see how the value of z (replaced by bold z[c] in the main marginalized expression) has an influence on defining that distribution, since the parameters of the Gaussian are constants even though z is supposed to represent the mean for the Gaussian.

It is also unclear to me why we need c to be in the shape of that funsor since z is already in the R^3 shape. It would make more sense to me to receive c if the funsor were also indexed by bold z of shape R^{2 x 3} since that would require knowing which component we are talking about.

Still about that Gaussian, how does it “know” x is the dimension for the Gaussian-distributed variable, and not c or z?

Another thing that puzzled me was that a normalized Gaussian is a product of a tensor and a Gaussian funsor. How is the Gaussian funsor not normalized by its own definition?