Commit 1f01a3d5 by Ying-Qiu Zheng

### Update 2021JUL21.md

parent e1ac92f9
 ... ... @@ -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 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}} \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}))^{y_{nk}}  The marginal distribution of $\mathbf{x}_{n}^{L}, \mathbf{x}_{n}^{H}$ is ... ... @@ -15,6 +15,10 @@ The marginal distribution of $\mathbf{x}_{n}^{L}, \mathbf{x}_{n}^{H}$ is \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})  In summary, in addition to finding the the hyper-parameters $\pi, \mu, \Sigma_{k}^{H}, \Sigma^{L}_{k}$, we want to estimate a transformation matrix $\mathbf{U}$ such that $\mathbf{UX}^{H}$ is as close to $\mathbf{X}^{L}`$ as possible (or vice versa). ### Simulation results ... ...
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