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EM Algorithm for Mixtures of Lines [3 P]

Assume that the training examples $ {\bf x}_n \in \mathbb{R}^2$ with $ n=1,...,N$ were generated from a mixture of $ K$ lines

$\displaystyle P(x_{n,2} \vert z_{n,k}=1)$ $\displaystyle =$ $\displaystyle \mathcal{N}( x_{n,2} \vert \theta_{k,1} x_{n,1} + \theta_{k,2},\sigma_k)$ (7)

$\displaystyle \mathcal{N}( x \vert \mu,\sigma)$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi} \sigma} \exp \left( -\frac{(x-\mu)^2}{2 \sigma^2}\right)$ (8)

and the hidden variable $ z_{n,k}=1$ if $ {\bf x}_n$ is generated from line $ k$ and 0 otherwise. Derive the update equations for the M-step of the EM algorithm for the variables $ {\bf\theta}_{k}$ and $ \sigma_k$ .

Haeusler Stefan 2011-12-06