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Let us assume that I am working on a dataset of black and white dog images.

Each image is of size $28 \times 28$.

Now, I can say that I have a sample space $S$ of all possible images. And $p_{data}$ is the probability distribution for dog images. It is easy to understand that all other images get a probability value of zero. And it is obvious that $n(S)= 2^{28 \times 28}$.

Now, I am going to design a generative model that sample from $S$ using $p_{data}$ rather than random sampling.

My generative model is a neural network that takes random noise (say, of length 100) and generates an image of the size $28 \times 28$. My function is learning a function $f$, which is totally different from the function $p_{data}$. It is because of the reason that $f$ is from $R^{100}$ to $S$ and $p_{data}$ is from $S$ to $[0,1]$.

In the literature, I often read the phrases that our generative model learned $p_{data}$ or our goal is to get $p_{data}$, etc., but in fact, they are trying to learn $f$, which just obeys $p_{data}$ while giving its output.

Am I going wrong anywhere or the usage in literature is somewhat random?

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You're right! The generative model $f$ is not the same as the probability density (p.d.f.) function $p_{data}$. The kind of phrases you've referred to are to be interpreted informally. You learn $f$ with the hope that sampling a latent vector $z$ from some known distribution (from which it is easy to sample), results in $f(z)$ that has the probability density function $p_{data}$. However, merely learning $f$ does not give you the power to estimate what $p_{data}(x)$ is for some image $x$. Learning $f$ only gives you the power to sample according to $p_{data}(\cdot)$ (if you've learned an accurate such $f$).

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