Optimize res_beta_binom()

[joethorley]

Currently runs very slowly. Consider using C++ code.

[nehill197]

I wrote a log-likelihood function in C++ to speed up the optimization routine (#66).
We decided to try improving the R code itself before adding the dependency to Rcpp.

First we added a scalar case to the log-likelihood function, which improves the speed for the scalar case required by stats::optimize().

As this was still slower than we would like, we next implemented a function to optimize a vector of all the probability values required for the deviance calculation at the same time.

Lastly, we added memoization to the log-likelihood function for the lgamma() calls that are functions of the data (x and size, which stay constant).

Based on the code below, we chose n = 800 as the criterion to use the memoized version of the function.

x <- 1:1000 + 0.5

lgamma3 <- function(x) {
  lgamma(x) + lgamma(x + 1) + lgamma(x + 2)
}

lgamma3_mem <- memoise::memoise(lgamma3)

bench::mark(
  for (i in 1:250) lgamma(x) + lgamma(x + 1) + lgamma(x + 2),
  for (i in 1:250) lgamma3_mem(x)
)