Exercise 17.4 - EM for HMMs with tied mixtures

Answers

In this case on term in the auxiliary function reads:

\begin{aligned} \log p (𝐳_{1 : T}, 𝐱_{1 : T}) & = \sum_{j = 1}^{J} 𝕀 [z_{1} = j] \cdot \log π_{j} + \sum_{t = 2}^{T} \sum_{i = 1}^{J} \sum_{j = 1}^{J} 𝕀 [z_{t - 1} = i, z_{t} = j] \cdot \log 𝐀_{i, j} \\ + \sum_{t = 1}^{T} \sum_{j = 1}^{J} 𝕀 [z_{t} = j] \cdot \log (\sum_{k = 1}^{K} w_{𝑗𝑘} \cdot 𝒩 (𝐱_{t} | μ_{k}, Σ_{k})) . \end{aligned}

Following the same method as in exercise 17.4, we build an internal auxiliary function with new latent variables so it reads:

Q^{I} (𝜃, 𝜃^{old}) = \sum_{n, t} \sum_{j, k} 𝔼 [𝕀 [h_{n, t, j, k} = 1]] \cdot \log 𝒩 (𝐱 | μ_{k}, Σ_{k}),

what left is the same as in an ordinary GMM model.

solour_lfq

2021-03-24 13:42