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Computing Mixture Models with EM

  • Given n objects O = {o1, …, on}, we want to mine a set of parameters Θ = {θ1, …, θk} s.t.,P(O|Θ) is maximized, where θj = (μj, σj) are the mean and standard deviation of the j-th univariate Gaussian distribution
  • We initially assign random values to parameters θj, then iteratively conduct the E- and M- steps until converge or sufficiently small change
  • At the E-step, for each object oi, calculate the probability that oi belongs to each distribution,

\[P(\theta_{j} |o_{i},\theta)=\frac{P(o_{i}|\theta_{j})}{\sum_{l=1}^{k}P(o_{i} |\theta_{l})}\]

  • At the M-step, adjust the parameters θj = (μj, σj) so that the expected likelihood P(O|Θ) is maximized

\[\mu_{j}=\sum_{i=1}^{n}o_{i}\frac{P(\theta_{j} |o_{i},\theta)}{\sum_{l=1}^{n}P(\theta_{j} |o_{l},\theta)}=\frac{{\sum_{i=1}^{n}o_{i}P(\theta_{j} |o_{i},\theta)}}{{\sum_{i=1}^{n}P(\theta_{j} |o_{i},\theta)}}\]

\[\sigma_{j}=\sqrt{\frac{\sum_{i=1}^{n}P(\theta_{j} |o_{i},\theta)(o_{i}-\mu_ {j})^{2}}{\sum_{i=1}^{n}P(\theta_{j} |o_{i},\theta)}}\]

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