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Generative Model
- Given a set of 1-D points X = {x1, …, xn} for clustering analysis & assuming they are generated by a Gaussian distribution:
\[N(\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\]
- The probability that a point xi ∈ X is generated by the model
\[P(x_{i}|\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x_{i}-\mu)^{2}}{2\sigma^{2}}}\]
- The likelihood that X is generated by the model:
\[L(N(\mu,\sigma^{2}):X)=P(X|\mu,\sigma^{2})=\prod_{i=1}^{n} \frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x_{i}-\mu)^{2}}{2\sigma^{2}}}\]
- The task of learning the generative model: find the parameters μ and σ2 such that
\[N(\mu_{o},\sigma_{o}^{2})=argmax{L(N(\mu,\sigma^{2}):X)}\]
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