逻辑回归“概率”函数（不是有效的 pdf ......）

Question

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The idea behind logistic regression is to estimate the posterior class conditional probability, given observation x for class C_k , with a sigmoid f(C_k| x)=1/(1+exp(-w*x)) to compute the weights vector w .

In every book I've read (eg, Bishop's PRML) f(C_k| x) is a probability density function but this is definitely not a valid pdf since the integral from minus infinity to infinity does not equal to 1 (nor it could be by any normalization since the integral is infinite).

Appreciate any explanations in this matter

Answer 1

You got it wrong, there is not integral from -inf to +inf. It is a discrete distribution p(c_k | x), and in case of logistic regression you have two classes c=1 and c=0. The model outputs the probability of belonging to class c=1. If you subtract p(c = 1 | x) from 1, you get the probability of the other class: p(c = 0 | x) = 1 - p(c = 1 | x). Softmax regression extends this to more than two classes by applying the softmax instead of the sigmoid or logistic function.