### update

parent 3b366a5b
 ## Fusion of High-quality and low-quality classification models ### Graphical models ![diagram1](/figs/2021JUL01/diagram-20210630-2.png) ### (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. ... ... @@ -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.
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