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Model-Based Clustering

  • A set C of k probabilistic clusters C1, …,Ck with probability density functions f1, …, fk, respectively, and their probabilities ω1, …, ωk.
  • Probability of an object o generated by cluster Cj is

\[P(o|C_{j})=w_{j}f_{j}(o)\]

  • Probability of o generated by the set of cluster C is

\[P(o|C)=\sum_{j=1}^{k}w_{j}f_{j}(o)\]

  • Since objects are assumed to be generated independently, for a data set D = {o1, …, on}, we have,

\[P(D|C)=\prod_{i=1}^{n}P(o_{i}|C)=\prod_{i=1}^{n}\sum_{j=1}^{k}w_{j}f_{j}(o_{i})\]

  • Task: Find a set C of k probabilistic clusters s.t. P(D|C) is maximized
  • However, maximizing P(D|C) is often intractable since the probability density function of a cluster can take an arbitrarily complicated form
  • To make it computationally feasible (as a compromise), assume the probability density functions being some parameterized distributions

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