  A Hilbert space embedding for probability measures has recently been
proposed, with applications including dimensionality reduction, homogeneity
testing, and independence testing. This embedding represents any probability
measure as a mean element in a reproducing kernel Hilbert space (RKHS). A
pseudometric on the space of probability measures can be defined as the
distance between distribution embeddings: we denote this as $\gamma_k$, indexed
by the kernel function $k$ that defines the inner product in the RKHS.
  We present three theoretical properties of $\gamma_k$. First, we consider the
question of determining the conditions on the kernel $k$ for which $\gamma_k$
is a metric: such $k$ are denoted {\em characteristic kernels}. Unlike
pseudometrics, a metric is zero only when two distributions coincide, thus
ensuring the RKHS embedding maps all distributions uniquely (i.e., the
embedding is injective). While previously published conditions may apply only
in restricted circumstances (e.g. on compact domains), and are difficult to
check, our conditions are straightforward and intuitive: bounded continuous
strictly positive definite kernels are characteristic. Alternatively, if a
bounded continuous kernel is translation-invariant on $\bb{R}^d$, then it is
characteristic if and only if the support of its Fourier transform is the
entire $\bb{R}^d$. Second, we show that there exist distinct distributions that
are arbitrarily close in $\gamma_k$. Third, to understand the nature of the
topology induced by $\gamma_k$, we relate $\gamma_k$ to other popular metrics
on probability measures, and present conditions on the kernel $k$ under which
$\gamma_k$ metrizes the weak topology.
