Commit e1ac92f9 authored by Ying-Qiu Zheng's avatar Ying-Qiu Zheng
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Update 2021JUL21.md

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### Model formulation
Suppose $`\mathbf{X}^{H}, \mathbf{X}^{L}`$ are $`N \times V`$ feature matrices (e.g. connectivity between $`N`$ thalamus voxels and $`V`$ whole brain voxels). Note that they can have different dimensions in practice. To keep notations uncluttered, we suppose the number of voxels in high- and low-quality images are the same for a given subject. Now we assume $`\mathbf{X}^{H}, \mathbf{X}^{L}`$ share the same latent variable $`Y`$, which is a $`N \times K`$ binary matrix representing the voxels' classes.
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
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 log-likelihood can be written as
```math
\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}(\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n}|\mu_{k},\Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n}|\mu_{k},\Sigma_{k}^{H}))^{y_{nk}}
```
The marginal distribution of $`\mathbf{x}_{n}^{L}, \mathbf{x}_{n}^{H}`$ is
```math
\log p(\mathbf{x}^{H}_{n}, \mathbf{x}^{L}_{n} | \mathbf{U},...\mathbf{\pi}, \mathbf{\mu}_{k},...\mathbf{\Sigma}^{H}_{k},...\mathbf{\Sigma}^{L}_{k},...)=\sum_{k=1}^{K}\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n}|\mu_{k},\Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n}|\mu_{k},\Sigma_{k}^{H})
```
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