[WIP] fit2: fast version of fitting algorithm

[ajinkya-k]

would complete #13 when done

started working on a faster version of the fitting algorithm in this draft PR. Suggestions would be helpful!

I am thinking of experimenting with the utbl/ytbl stuff that @dmbates wrote about in the Embrace Uncertainty book.

link to the branch on my fork for easy access: https://github.com/ajinkya-k/BetaRegression.jl/tree/ahk/fast-fit

TODOs:

• compare performance on a battery of tests
• pre-allocate all intermediate quantities like utbl/ytbl in EmbraceUncertainty

[ajinkya-k]

As I mentioned in #13, this leads to a decent speedup as is. I think the speed up will improve once I allocate only once all vectors for intermediate quantities (like digamma and trigamma and mu_i) that are used in both the score and the information matrix using @dmbates 's utbl idea

julia> @be fit!(bm)
Benchmark: 2517 samples with 1 evaluation
 min    31.459 μs (242 allocs: 31.844 KiB)
 median 33.042 μs (242 allocs: 31.844 KiB)
 mean   37.471 μs (242 allocs: 31.844 KiB, 0.19% gc time)
 max    2.274 ms (242 allocs: 31.844 KiB, 97.26% gc time)

julia> @be BetaRegression.fit2!(bm)
Benchmark: 3146 samples with 1 evaluation
 min    26.958 μs (172 allocs: 27.406 KiB)
 median 27.584 μs (172 allocs: 27.406 KiB)
 mean   28.967 μs (172 allocs: 27.406 KiB, 0.15% gc time)
 max    1.844 ms (172 allocs: 27.406 KiB, 97.55% gc time)