# Why do we use $D(x \mid y)$ and not $D(x,y)$ in conditional generative adversarial networks?

In conditional generative adversarial networks (GAN), the objective function (of a two-player minimax game) would be

$$\min _{G} \max _{D} V(D, G)=\mathbb{E}_{\boldsymbol{x} \sim p_{\text {data }}(\boldsymbol{x})}[\log D(\boldsymbol{x} | \boldsymbol{y})]+\mathbb{E}_{\boldsymbol{z} \sim p_{\boldsymbol{z}}(\boldsymbol{z})}[\log (1-D(G(\boldsymbol{z} | \boldsymbol{y})))]$$

The discriminator and generator both take $$y$$, the auxiliary information.

I am confused as to what will be the difference by using $$\log D(x,y)$$ and $$\log(1-D(G(z,y))$$, as $$y$$ goes in input to $$D$$ and $$G$$ in addition to $$x$$ and $$z$$?

The joint probability $$D(x,y)$$ is the probability of x and y both happening together.
The conditional probability $$D(x | y)$$ is the probability that x happens, given that y has already happened. So, $$D(x,y) = D(y) * D(x | y)$$.
Notice that, in a C-GAN, we have some extra information that is given, like a class label $y$. We actually don't care at all about how likely that information is to appear. We care only about how likely it is to appear with a given $x$ from the source distribution, versus how likely it is to appear with a given $z$ from the generated distribution.
If you tried to minimize the joint probabilities, you would be attempting to change something that the networks have no ability to control (the chance of $y$ appearing).