Consider the one-dimensional logistic model
P(y=1∣x;w)=σ(wx)=11+e−wx.P(y=1\mid x;w) = \sigma(wx) = \frac{1}{1+e^{-wx}}.P(y=1∣x;w)=σ(wx)=1+e−wx1.
Given the i.i.d. observations (x1,y1)=(1,1)(x_1,y_1)=(1,1)(x1,y1)=(1,1) and (x2,y2)=(−1,1)(x_2,y_2)=(-1,1)(x, derive the maximum likelihood estimate .