Exercise 11.7 - Manual calculation of the M step for a GMM

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For question (a), we are to optimize:

\sum_{n = 1}^{3} \sum_{k = 1}^{2} r_{𝑛𝑘} (\log π_{i} + \log 𝒩 (x_{n} | μ_{k}, σ_{k}^{2})) .

For question (b), the new optimal assignment for $π_{1}$ , $π_{2}$ is derived by differentiating the auxiliary function added with a regularizer to ensure $π_{1} + π_{2} = 1$ :

\frac{∂𝑄 + λ (π_{1} + π_{2} - 1)}{\partial π_{1}} = \frac{\sum_{n = 1}^{3} r_{n, 1}}{π_{1}} + λ = \frac{1.4}{π_{1}} + λ,

\frac{∂𝑄 + λ (π_{1} + π_{2} - 1)}{\partial π_{2}} = \frac{\sum_{n = 1}^{3} r_{n, 2}}{π_{2}} + λ = \frac{1.6}{π_{2}} + λ .

Setting both gradients to zero ends up with:

\begin{aligned} π_{1} & = \frac{7}{15}, \\ π_{2} & = \frac{8}{15}, \\ λ & = - 3 . \end{aligned}

For question (c), we use the results from exercise 11.2:

\begin{aligned} μ_{1} & = \frac{\sum_{n = 1}^{3} r_{n 1} \cdot x_{n}}{\sum_{n = 1}^{3} r_{n 1}} = \frac{25}{7}, \\ μ_{2} & = \frac{\sum_{n = 1}^{3} r_{n 2} \cdot x_{n}}{\sum_{n = 1}^{3} r_{n 2}} = \frac{65}{4} . \end{aligned}

solour_lfq

2021-03-24 13:42

Exercise 11.7 - Manual calculation of the M step for a GMM

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