replace filldist with product_distribution

[yebai]

Mooncake's performances are consistently better after switching to product_distribution from filldist / arraydist.

Before the change (results on main)

After the change

[yebai]

@penelopeysm, the model is not optimised for performance, but removes some heuristics from the early days of Julia and Turing

[yebai]

This is ready for reviewing. Mooncake's performance is generally better with product_distribution, which is nice.

[penelopeysm]

Thanks Hong. I'm mildly concerned about the use of product_distribution(fill(...)) over filldist. The latter is generally more performant because it uses lazy arrays and so even though the grad / primal ratio looks better, I wonder if it is masking some sins. In general from the comparison you posted here, it seems that Mooncake does much better on strict arrays rather than lazy arrays so I wonder if there is some performance optimisation on lazy arrays that will help to bring the performance down on filldist. Many people still use filldist (indeed it's pretty much everywhere on our docs page / code examples out there) so I think it's not unreasonable for people to want Mooncake to perform well on it.

In any case I don't consider this to be a problem for ADTests because the model definition is shown on the website so all this information is given upfront, so will merge.