Exercise 11.4 - EM for mixture of Student distributions

Answers

The log-likelihood for the complete data set for the mixture of Student distribution is:

l_{c} (𝐱, z_{k} = 1) = π_{k} \cdot \frac{Γ (\frac{D}{2} + \frac{v_{k}}{2})}{Γ (\frac{v_{k}}{2})} \cdot \frac{| Σ_{k} |^{- \frac{1}{2}}}{v_{k}^{\frac{D}{2}} π^{\frac{D}{2}}} \times {(1 + \frac{1}{v_{k}} (𝐱 - μ) Σ_{k}^{- 1} (𝐱 - μ))}^{- \frac{v_{k} + D}{2}} .

Following the similar path as exercise 11.2 and exercise 11.3, we have:

Q (𝜃, 𝜃^{old}) = \sum_{n = 1}^{N} \sum_{k = 1}^{K} r_{𝑛𝑘} \cdot (\log π_{k} + \log (\frac{Γ (\frac{D}{2} + \frac{v_{k}}{2})}{Γ (\frac{v_{k}}{2})} \cdot \frac{| Σ_{k} |^{- \frac{1}{2}}}{v_{k}^{\frac{D}{2}} π^{\frac{D}{2}}} \times {(1 + \frac{1}{v_{k}} (𝐱 - μ) Σ_{k}^{- 1} (𝐱 - μ))}^{- \frac{v_{k} + D}{2}})) .

The rest is ordinary EM, whose detail symbolic forms is too complex to write down explicitly.

solour_lfq

2021-03-24 13:42