Question regarding notation in Goodfellow's GAN paper On page 3 of the GAN paper in the caption of Figure 1 there is this expression: 
$D^{*}(x) = \frac{p_{data}(x)}{p_{data}(x) + p_{g}(x)}$ 
where $p_{data}$ is the distribution of the training data and $p_{g}$ is the distribution of the generator. 
What is the difference between $p_{data}(x)$ and $p_{data}$ ? 
Does $D^{*}(x)$ equate to a scalar?
Thanks! 
 A: As you correctly note, $p_{data}$ is the distribution of the training data and $p_{g}$ is the distribution of the generator. In addition, as the authors note: "$D(x)$ represents the probability that $x$ came from the data rather than $p_g$". 
OK, let's clarify something here: $D$ is a discriminator, it is defined to be a multilayer perceptron (page 2) but it can understood more easily as being a probabilistic classifier. Thus $D$ defines a distribution and $D(x)$ defines a sample from that distribution based on the input arguments $x$. Similarly $p_{data}$ is the distribution of the training data and $p_{data}(x)$ is the probability of sample $x$ coming from $p_{data}$.
Coming to $D^*$ now, I hope it is clear that this defines another distribution (and according to authors what is an optimal discriminator). Therefore, $D^*(x)$ is a sample from that distribution based on the input arguments $x$. So, yes, you are correct that $D^*(x)$ can equate to a scalar (it $x$ holds a single sample) but in most cases is a vector.
