Question 1

Figure from the original question paper **NOTE:** Enter the answer correct to one decimal place Consider a dataset of 200 points where the $i$-th data point is given by: $$\mathbf{x}_i = a_i \cdot \begin{bmatrix} 3 \\ 1 \\ 0 \\ 2 \end{bmatrix} + b_i \cdot \begin{bmatrix} 1 \\ 0 \\ 3 \\ 0 \end{bmatrix},$$ where $a_i$ and $b_i$ are real numbers such that $$\sum_{i=1}^{200} a_i = \sum_{i=1}^{200} b_i = 0.$$ Standard PCA is performed on this dataset. If the top two principal components are retained and used to reconstruct the dataset, what is the reconstruction error? A train running between two stations A and B will be late on any day by a random amount $X$, where $X \sim \text{Uniform}[0, \theta]$. Suppose the train is late by random amounts (in minutes) $$10, 45, 17, 20, 49, 52, 15, 4, 32, 35$$ independently on 10 days. Consider a $\text{Uniform}[0, 60]$ prior for the parameter $\theta$. Find the maximum aposteriori (MAP) estimate of $\theta$. Recall that $\hat{\theta}_{MAP} = \underset{\theta}{\arg\max}\ f(\theta \mid \{X_1, \ldots, X_n\})$, where $f$ is the posterior distribution.