Let $p(x)$ denote a probability density function defined over the random variable $x \in R^{m}$. Given an arbitrary feature map $\phi : R^{m} \rightarrow \textit{R}$, we can represent the density $p(x)$ based on its expected value under this feature map: $\mu_{x}=\int_{R^{m}} \phi (x)p(x)dx$.

The key idea with Hilbert space embeddings of distributions is that this equation will be injective, as long as a suitable feature map $\phi$ is used. This means that $\mu_{x}$ can serve as a sufficient statistic for $p(x)$, and any computations we want to perform on $p(x)$ can be equivalently represented as functions of the embedding $\mu_{x}$.