### 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).

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

- Here we adopt the Relevance Vector Machine (RVM) with ARD prior to find $`\mathbf{w}`$. Suppose $`\mathbf{w}`$ has a prior distribution $`\mathcal{N}(\mathbf{0}, \text{diag}(\alpha_{i}^{-1}))`$. We hope $`\alpha_{i}`$ is driven to Inf, if the associated feature is useless for prediction.

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@@ -87,7 +87,13 @@ And we compared three methods:

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

### 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.

#### On the high quality data

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. [ref](https://proceedings.neurips.cc/paper/2014/file/f9a40a4780f5e1306c46f1c8daecee3b-Paper.pdf)

#### On the low quality data

The low-quality coefficients have similar structured ARD priors (exp of a Gaussian Process) that may not share the same hyperparameters with the high-quality coefficients' priors. We seek to solve the hyperparameters for the low-quality classification model, marginalising over the posteriors of the high-quality model.

### 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.

#### On the high quality data

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. [ref](https://ohbm.sparklespace.net/srh-2591/)

#### On the low quality data

The low quality coefficients have similar priors to enforce sparsity.