Logistic regression estimating equations 2. (6 points) When yi is a binary 01 variable, it is awkward to model it as a linear function of 231-, since it is bounded: any line we choose, as long as it has nonzero slope, will eventually leave the interval [0,1]. A more natural choice is to estimate some linear function and then pass it through a nonlinearity. For various reasons, the most popular choice of nonlinearity is the logistic function e2 1 =1+ez 21+e'z' f(Z) This is also sometimes called a sigmoid function because if the plot looks a bit like the letter S. A (simple) logistic regression is a regression of a binary response 3; on a single variable 3:, where we estimate g) 2 ex + ban), for f as dened above. Note that because 3; is always 0 or 1, it may seem nonsensical to predict it to be, egg 0.7. It does, however, make sense to predict that it has a 70% probability of being 1 and a 30% probability of being zero. We could try to use the squared error loss or absolute loss to estimate a simple logistic regression, but it is more natural to use the og-likelihood, i.e. the probability that y1,...,yn would come out exactly the same way they did if we were to collect ii new data points. If yl, . . . , yn were independent, and each yi had a probability exactly 3}:- of turning out to be 1, then the corresponding loglikelihood loss (or log loss for short) is L( n)_ log(g}) ify=1 9'9 _ mgr3}) ify:0 (a) (3 points) If we use the model 3) : f(a + bar), show that the logloss can be written as L(y1: f(0u 'l EYE\") : 91(61 + (5131') _1Og(1+ 83+531)