Multinomial probabilities

Topic: Logistic regression · more questions on this topic
‹ Linear discriminant analysis implies a logistic model Complete separation ›

With three classes and class 1 as the reference, suppose at some x that \beta_2^\top x=0.5 and \beta_3^\top x=-0.2. Find \tau_1, \tau_2, \tau_3 and the log odds of class 3 against class 2.

Worked answer (click to show)

The common denominator is

1+e^{0.5}+e^{-0.2}=1+1.6487+0.8187=3.4674.

So \tau_1=1/3.4674=0.288, \tau_2=1.6487/3.4674=0.476, and \tau_3=0.8187/3.4674=0.236, which sum to 1.000. The log odds of class 3 against class 2 are

\log\frac{\tau_3}{\tau_2}=(\beta_3-\beta_2)^\top x=-0.2-0.5=-0.7.

The common denominator cancels in the pairwise comparison.