  We introduce Isolation Distributional Kernel as a new way to measure the
similarity between two distributions. Existing approaches based on kernel mean
embedding, which convert a point kernel to a distributional kernel, have two
key issues: the point kernel employed has a feature map with intractable
dimensionality; and it is {\em data independent}. This paper shows that
Isolation Distributional Kernel (IDK), which is based on a {\em data dependent}
point kernel, addresses both key issues. We demonstrate IDK's efficacy and
efficiency as a new tool for kernel based anomaly detection for both point and
group anomalies. Without explicit learning, using IDK alone outperforms
existing kernel based point anomaly detector OCSVM and other kernel mean
embedding methods that rely on Gaussian kernel. For group anomaly detection,we
introduce an IDK based detector called IDK$^2$. It reformulates the problem of
group anomaly detection in input space into the problem of point anomaly
detection in Hilbert space, without the need for learning. IDK$^2$ runs orders
of magnitude faster than group anomaly detector OCSMM.We reveal for the first
time that an effective kernel based anomaly detector based on kernel mean
embedding must employ a characteristic kernel which is data dependent.
