Commit 95a599c9 authored by Ying-Qiu Zheng's avatar Ying-Qiu Zheng
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Update 2021JUL21.md

parent 9aba6762
......@@ -7,7 +7,7 @@ Suppose $`\mathbf{X}^{H}, \mathbf{X}^{L}`$ are $`N \times V`$ feature matrices (
For a single voxel, suppose $`\mathbf{y}_{n} \sim \text{multinomial}(\mathcal{\pi})`$, and $`p(\mathbf{x}^{L}_{n}|y_{nk}=1) = \mathcal{N}(\mu_{k}, \Sigma_{k}^{L})`$. To use high-quality data to inform the inference on low-quality data, we assume $`p(\mathbf{x}^{H}_{n}|y_{nk}=1, \mathbf{U}) = \mathcal{N}(\mathbf{U}\mathbf{x}^{H}_{n}|\mu_{k}, \Sigma_{k}^{H})`$ where $`\mathbf{U}^{T}\mathbf{U} = \mathbf{I}`$. The complete likelihood can be written as
```math
p(\mathbf{x}^{H}_{n}, \mathbf{x}^{L}_{n}, \mathbf{y}_{n}|\mathbf{U}, \mathbf{\pi}, \mathbf{\mu}_{k},...\mathbf{\Sigma}^{H}_{k},...\mathbf{\Sigma}^{L}_{k},...)
\log p(\mathbf{x}^{H}_{n}, \mathbf{x}^{L}_{n}, \mathbf{y}_{n}|\mathbf{U},...\mathbf{\pi}, \mathbf{\mu}_{k},...\mathbf{\Sigma}^{H}_{k},...\mathbf{\Sigma}^{L}_{k},...) = \prod_{k=1}^{K}(\mathcal{N}(\mathbf{x}^{L}_{n}|\mu_{k},\Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n}|\mu_{k},\Sigma_{k}^{H}))^{z_{nk}}
```
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