Question 1

Kernel regression with a polynomial kernel is applied on the following dataset with two features: $$X = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, \qquad y = [2, 1, 2]^T$$ Weight vector can be written as $w = \phi(X)\alpha$, where $\phi$ is the transformation mapping corresponding to the kernel $k(x_i, x_j) = (1 + x_i^T x_j)^2$. The vector $\alpha$ is given by $(K)^{-1}y$, where $K$ is the kernel matrix. Based on the above data, answer the given subquestions. Figure from the original question paper Kernel regression with a polynomial kernel is applied on the following dataset with two features: $$X = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, \qquad y = [2, 1, 2]^T$$ Weight vector can be written as $w = \phi(X)\alpha$, where $\phi$ is the transformation mapping corresponding to the kernel $k(x_i, x_j) = (1 + x_i^T x_j)^2$. The vector $\alpha$ is given by $(K)^{-1}y$, where $K$ is the kernel matrix. Based on the above data, answer the given subquestions. Figure from the original question paper A binary classification dataset has 2000 data points belonging to $\{0,1\}^2$. A Naive Bayes algorithm was run on the same dataset, resulting in the following estimates: $$\begin{aligned} \hat{p}, &\text{ estimate for } P(y = 1) = 0.4 \\ \hat{p}_1^0, &\text{ estimate for } P(f_1 = 1 \mid y = 0) = 0.25 \\ \hat{p}_2^0, &\text{ estimate for } P(f_2 = 1 \mid y = 0) = 0.35 \\ \hat{p}_1^1, &\text{ estimate for } P(f_1 = 1 \mid y = 1) = 0.15 \\ \hat{p}_2^1, &\text{ estimate for } P(f_2 = 1 \mid y = 1) = 0.05 \end{aligned}$$ Based on the above data, answer the given subquestions. Figure from the original question paper