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### Comments to solution for OASI Problem 1.3 ###
### Frank Miller 2020-10-13                   ###
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# The function g to be maximised and gradient
# The data can be options of the function like here or just defined as global variables
# To have it as options allows to call the function also for a part of the data (one observation or mini-batch)
g <- function(b,x,y)
{
  # to ensure that h1<1, we add a little 1e-9; otherwise log(1-h1) might crash
  h1 <- 1/(1+exp(-b[1]-b[2]*x)+1e-9)
  sum(log(h1)*y + log(1-h1)*(1-y))
}
gradient <- function(b,x,y)
{
  h1  <- 1/(1+exp(-b[1]-b[2]*x))
  dg1 <- sum((y - h1))
  dg2 <- sum((y - h1)*x)
  c(dg1,dg2)
}

# Data
xobs <- c(0,0,0, 0.1,0.1, 0.3,0.3, 0.9,0.9,0.9)
yobs <- c(0,0,1,   0,  1,   1,  1,   0,  1,  1)

# call functions e.g. with g(c(-0.2,1),xobs,yobs)  or  gradient(c(-0.2,1),xobs,yobs)  where c(-0.2,1) is the point of evaluation