#prediction-variance

Question 1: Single choice

hardAI-checkedLinear Models & Regularization

In $y = wx + \epsilon$ with $\epsilon \sim \mathcal{N}(0, \sigma^2 x^2)$ and $x \in [1, 4]$ , the ML estimate from $n$ points is $\hat w_n = \frac{1}{n}\sum_i y_i/x_i$ , so the prediction $\hat w_n x$ has variance $x^2\sigma^2/n$ , largest at $x = 4$ . You may query one more input $x_{n+1}$ . Which choice reduces this largest prediction variance the most?

Query $x_{n+1} = 4$ , where the prediction variance is largest1
Query $x_{n+1} = 1$ , where the noise on $y$ is smallest2
Every choice of $x_{n+1}$ in $[1, 4]$ reduces it by the same amount3
Query $x_{n+1}$ uniformly at random, which beats any fixed choice on average4

#prediction-variance

Timed test