Current Slide

• 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

\[N(\mu_{o},\sigma_{o}^{2})=argmax{L(N(\mu,\sigma^{2}):X)}\]