How discriminator output in GAN is a probability distribution?

Recently I asked a question about GAN,What is the intuition behind the expected value in orginal GAN papers objective function? , In there I came to know that the discriminator output is viewed as a probability distribution. Although I have an amateurish knowledge in maths, I have created many vanilla GAN models. In those models, discriminator neural network has one output node which yields a scaler value between 0 and 1 for sigmoid activation function.

Now lets consider this pseudo example, lets say I have a batch of three real images $$x = \left \{ x_1,x_2,x_3\right \}$$ and we are passing this batch to our discriminator $$D(x)$$, lets assume these are its corresponding outputs $$y = \left \{ 0.1,0.8,0.5\right \}$$ given the last layer activation is $$sigmoid$$.

One of the rule of probability distribution is sum of all the probabilities or area under the curve should be equal to one but if we take sum of $$0.1+0.8+0.5= 1.4$$ we get a sum more than one.

Here we are presuming that the output activation is $$sigmoid$$ but I have also seen and implemented many vanilla GAN models where the last activation function is $$tanh$$ ,that is, $$D(x)= (-1,1)$$.

Given this, how discriminator output in GAN is considered as a probability distribution?

• I think there is a misunderstanding in "probabilities must sum to 1". The output on the sigmoid is a probability distribution and does sum to 1. The output of D is equivalent to P(image is Generated by G). For x1, 0.1+0.9 =1, except that you never output 0.9 because the value is obvious. You are summing probabilities over the different images, which is just not something you can do. Each image has a probability to be fake but the sum of these 3 images' probabilities is not a quantity you can claim to sum to 1 – Romain Mar 27 at 11:09