MMiDS 6.5: Self-Assessment Quiz

Which of the following is the Schur complement of the block \(B_{11}\) in the positive definite matrix \(B = \begin{pmatrix} B_{11} & B_{12} \\ B_{12}^T & B_{22} \end{pmatrix}\)?

• a) \(B_{22} - B_{12}^T B_{11}^{-1} B_{12}\)
• b) \(B_{11} - B_{12} B_{22}^{-1} B_{12}^T\)
• c) \(B_{22}\)
• d) \(B_{11}\)

Which of the following is true about the Schur complement \(B/B_{11}\) of the block \(B_{11}\) in a positive definite matrix \(B\)?

• a) It is always symmetric.
• b) It is always positive definite.
• c) Both a and b.
• d) Neither a nor b.

What is the conditional distribution of \(X_1\) given \(X_2\) in a multivariate Gaussian distribution?

• a) \(X_1 | X_2 \sim \mathcal{N}(\mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2), \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}^T)\)
• b) \(X_1 | X_2 \sim \mathcal{N}(\mu_1 - \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2), \Sigma_{11} + \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}^T)\)
• c) \(X_1 | X_2 \sim \mathcal{N}(\mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2), \Sigma_{11} + \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}^T)\)
• d) \(X_1 | X_2 \sim \mathcal{N}(\mu_1 - \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2), \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}^T)\)

In a linear-Gaussian system, which of the following is true about the observation process \(Y_t\)?

• a) It is a noisy linear function of the current state \(X_t\).
• b) It is a noisy linear function of the previous state \(X_{t-1}\).
• c) It is a noisy linear function of the next state \(X_{t+1}\).
• d) It is independent of the state \(X_t\).

In a linear-Gaussian system, which of the following is true about the conditional independence relationships?

• a) \(X_t\) is conditionally independent of \(Y_{1:t-1}\) given \(X_{t-1}\).
• b) \(Y_t\) is conditionally independent of \(Y_{1:t-1}\) given \(X_t\).
• c) \(X_t\) is conditionally independent of \(X_{t-2}\) given \(X_{t-1}\).
• d) All of the above.

In the location tracking example, what does the observation \(Y_t\) represent?

• a) The true position of the object.
• b) The noisy measurement of the object's position.
• c) The estimated position of the object based on the Kalman filter.
• d) The velocity of the object.

When applying the Kalman filter to location tracking, which of the following describes the state vector \(X_t\)?

• a) The observed position at time \(t\).
• b) The observed velocity at time \(t\).
• c) The true position and velocity at time \(t\).
• d) The measurement noise at time \(t\).

In the location tracking example, what does the observation matrix \(H\) represent?

• a) The mapping from the state (position and velocity) to the observed position.
• b) The mapping from the state (position and velocity) to the observed velocity.
• c) The mapping from the previous state to the current state.
• d) The mapping from the current state to the next state.

How is the innovation (or residual) \(e_t\) computed in the Kalman filter?

• a) \(e_t = Y_t - H\mu_{t-1}\)
• b) \(e_t = Y_t - H\mu_t\)
• c) \(e_t = Y_t - H\mu_{\text{pred}}\)
• d) \(e_t = Y_t - \mu_{\text{pred}}\)

In the provided Python implementation of the Kalman filter, what is the purpose of the line K = Sig_pred @ H.T @ Sinv?

• a) To compute the predicted state.
• b) To compute the innovation (prediction error).
• c) To compute the Kalman gain matrix.
• d) To update the state covariance matrix.

In the Kalman filter, what does the Kalman gain matrix \(K_t\) represent?

• a) The covariance matrix of the state estimate at time \(t\).
• b) The covariance matrix of the observation at time \(t\).
• c) The weight given to the observation at time \(t\) when updating the state estimate.
• d) The weight given to the previous state estimate when predicting the current state.