Question about mixture models intuition The following paragraph appears in Kevin P Murphy's 'Machine Learning A Probabilistic Perspective' chapter 'Mixture Models Discrete Hidden Variables'

Whilst there are some cases, such as the Gaussian Mixture Model
  discussed below, where there is a clear visual interpretation of the
  meaning of ‘cluster’, the reader should bear in mind that the
  intuitive meaning of ‘cluster’ is based on datapoints being ‘close’ in
  some sense to each other. The tremendous advantage of the realisation
  that mixture models generalise the idea of modelling clusters, is that
  we are freed from the conceptual constraint of distances in some
  Euclidean type sense. Instead, two datapoints become ‘close’ if they
  are both likely with respect to the model for that ‘cluster’. Hence,
  we can immediately start to ‘cluster’ all kinds of data – music,
  shopping purchases etc – things which do not necessarily have a
  natural ‘distance’ measure.

I did not quite understand the meaning of the statement "The tremendous advantage of the realisation
that mixture models generalise the idea of modelling clusters, is that we are
freed from the conceptual constraint of distances in some Euclidean type sense."
Would appreciate if someone explained this statement with an example to make it more easier to understand.
 A: Murphy means that "classical" cluster models (like K-means) are based on the concept of distance between clusters, but mixture models are based on probability instead, which can be more general. That's because distance is normally defined only between numbers or vectors, but probability may be modelled for objects of any nature.
For example, I could define and estimate a mixture model over pieces of DNA of length $n$. Let's assume that in the cluster $i$ the $j$th nucleotide in the sequence could be A, G, T or C with probabilities $p_i(A), p_i(G), p_i(T), p_i(C)$ independent of $j$. 
Then, given a set of sequences like ('ATTTTCA', 'GCCCGCGG', 'AGATCCCA', ...), you can fit your model (probabilities of each cluster and conditional probabilities of nucleotides given a cluster) to this set with EM algorithm. Thereafter, for each new sequence of DNA you could predict with this model, to which cluster this new sequence most probably belongs. 
Notice that to accomplish this task, you would never need to translate the DNA sequences into some vector space, nor calculate distances between them. And that's why mixture models are great.
