### (The most basic) model formualtion (panel (A))

### Panel A - (the most basic) model formulation (with classical ARD priors)

The model for high quality data classification follows a regression form with ARD priors. The low-quality model is trained on the posterior distribution of the high quality coefficients $`\mathbf{w}^{H}`$ to give a set of low-quality coefficients (with ARD priors likewise).

#### On the high quality data.

- Suppose $`\mathbf{X}^{H}`$ is the $`v\times d`$ feature matrix (e.g. connectivity profiles of $`v`$ voxels). $`\mathbf{t}`$ is the $`v\times 1`$ labels (0-1 variables). $`\mathbf{w}`$ is the $`d\times 1`$ coefficients, and $`\mathbf{y}=\sigma(\mathbf{X}^{H}\mathbf{w})`$ determines the probability for each class.

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@@ -86,12 +86,8 @@ And we compared three methods:

When $`d >> n`$, Lasso appears superior to the others.

### Panel B - structured ARD priors (in progress).

#### On the high quality data

Instead of $`\mathbf{w}\sim\mathcal{N}(0, \text{diag}(\alpha_1,...\alpha_d))`$, we assume the hyperparamters have a underlying structure, e.g., $`\mathbf{w}\sim\mathcal{N}(0, \text{diag}(\exp(\mathbf{u}))`$, where $`\mathbf{u}`$ is a Gaussian process $`\mathbf{u}\sim\mathcal{N}(\mathbf{0}, \mathbf{C}_{\Theta})`$ such that neighbouring features (i.e., adjoining/co-activating voxels) share similar sparsity.

#### On the low quality data

### Panel B - with structured ARD priors (in progress).

Instead of using ARD priors $`\mathbf{w}\sim\mathcal{N}(0, \text{diag}(\alpha_1,...\alpha_d))`$, we assume the hyperparamters have a underlying structure, e.g., $`\mathbf{w}\sim\mathcal{N}(0, \text{diag}(\exp(\mathbf{u}))`$, where $`\mathbf{u}`$ is a Gaussian process $`\mathbf{u}\sim\mathcal{N}(\mathbf{0}, \mathbf{C}_{\Theta})`$ such that neighbouring features (i.e., adjoining/co-activating voxels) share similar sparsity.

### Panel C - structured spike-and-slab priors (in progress).

#### On the high quality data

Instead of using ARD priors, we assume the coefficients have spike-and-slab priors with latent variables $`\gamma_i^{H}, i=1,2,...d`$, where $`\gamma_i\sim\text{Bernoulli}(\sigma(\theta))`$. The hyperparameter $`\theta`$ can be a Gaussian Process.

#### On the low quality data

### Panel C - with structured spike-and-slab priors (in progress).

Similarly, instead of using ARD priors, we assume the coefficients have spike-and-slab priors with latent variables $`\gamma_i^{H}, i=1,2,...d`$, where $`\gamma_i\sim\text{Bernoulli}(\sigma(\theta))`$. The hyperparameter $`\theta`$ can be a Gaussian Process.