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I am self-studying Kevin Murphy's book Machine learning - A probabilistic perspective and stumbled upon the following paragraph on biclustering. I understand the independence assumption - but am not yet seeing how this allows one to cluster. Can someone provide the intuition behind this?

enter image description here

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If I understand your question, the link between independence and clustering is this. That data points within each cluster are generated IID from a distribution belonging to the cluster.

Suppose we had a simple model, where each data point is assigned a colour token (red or blue) and a shape token (square or circle) and its colour-shape combination determines which probability distribution it is drawn from:

  1. red squares are IID Normal(0,1)
  2. blue squares are IID Normal(10,1)
  3. red circles are IID Normal(0,3)
  4. blue circles are IID Normal(10,3)

But when you get the data you generally aren't told which data points received which tokens. Or even how many different shapes / colours were involved. So if each combination is equally likely the resulting histogram of draws would look something like:

enter image description here

The inference process tries to recover this. The inference process he suggests isn't clear to me from the level of description given.

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