Question

1. Bias vs Variance in Ridge Regression: Consider the problem of estimating the real value parameter $\theta_0$ from the dataset $D$ with $N$ measurements: $y_i = \theta_0 + n_i; \quad n = 1, \dots, N$. (1) Here $n_i; i = 1, 2, \dots, N$ are independent zero-mean Gaussian noise samples of variance $\sigma^2$. Consider the ridge regression estimate specified by $\hat{\theta}_\lambda = \frac{\sum_{n=1}^{N} y_n}{N + \lambda}$ (2) (a) Show that the bias of this estimate is given by $\text{Bias} \left[ \hat{\theta}_{\text{ridge}} \right] = \frac{\lambda \theta_0}{N + \lambda}$. (3)

          1. Bias vs Variance in Ridge Regression: Consider the problem of estimating the real value parameter $\theta_0$ from the dataset $D$ with $N$ measurements:
$y_i = \theta_0 + n_i; \quad n = 1, \dots, N$.
(1)
Here $n_i; i = 1, 2, \dots, N$ are independent zero-mean Gaussian noise samples of variance $\sigma^2$. Consider the
ridge regression estimate specified by
$\hat{\theta}_\lambda = \frac{\sum_{n=1}^{N} y_n}{N + \lambda}$ (2)
(a) Show that the bias of this estimate is given by
$\text{Bias} \left[ \hat{\theta}_{\text{ridge}} \right] = \frac{\lambda \theta_0}{N + \lambda}$.
(3)

1. Bias vs Variance in Ridge Regression: Consider the problem of estimating the real value parameter θ0 from the dataset D with N measurements:
yi = θ0 + ni; n = 1, …, N.
(1)
Here ni; i = 1, 2, …, N are independent zero-mean Gaussian noise samples of variance σ^2. Consider the
ridge regression estimate specified by
θ̂λ = (∑n=1^N yn)/(N + λ) (2)
(a) Show that the bias of this estimate is given by
Bias[ θ̂ridge] = (λθ0)/(N + λ).
(3)

Added by Briana S.

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