A logistic rule in an unfamiliar setting

Topic: Logistic regression · more questions on this topic
‹ Reading the deviance

A fitted binary logistic model is

\log\frac{\mathbb{P}(Y=1\mid x)}{\mathbb{P}(Y=0\mid x)}=-3+0.8x_1-0.4x_2.

Assume Y=1 is the positive class.

  1. Find the probability at x=(4,2)^\top and give its classification under a probability threshold of 0.5.

  2. Give the decision-boundary equation and interpret the coefficient of x_1 with x_2 fixed.

  3. Explain two modelling differences between logistic regression and linear discriminant analysis.

  1. The linear predictor is -3+0.8(4)-0.4(2)=-3+3.2-0.8=-0.6. Thus

p=\frac{1}{1+e^{0.6}}=0.354,

and the prediction at threshold 0.5 is class 0.

  1. The boundary is -3+0.8x_1-0.4x_2=0. Increasing x_1 by one unit multiplies the conditional odds by e^{0.8}=2.226, holding x_2 fixed. It does not add 0.8 to the probability.

  2. Logistic regression specifies \mathbb{P}(Y\mid X) directly and does not require Gaussian class-conditional predictor distributions. Linear discriminant analysis specifies Gaussian X\mid Y with a shared covariance and class priors, then derives posterior probabilities. Both give linear log odds under these forms. Their assumptions and estimation procedures differ.