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?

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    $\begingroup$ 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 $\endgroup$ – Romain Mar 27 at 11:09

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