These questions are not due, they are just for your benefit. The TAs will go over the solutions in the lab session before the midterm.
Problem 1
Consider a discrete HMM with latent space of dimension K=2 and T=1 observations. The model has the following prior, transition matrix, and observation probability functions:
\begin{align} \boldsymbol{\pi} &= \begin{bmatrix} 0.8 \\ 0.2 \end{bmatrix} \\ \mathbf{A} &= \begin{bmatrix} 0.1 & 0.9 \\ 0.5 & 0.5 \end{bmatrix} \\ p(\mathbf{x}_1|\mathbf{z}_{1} = \begin{bmatrix} 1.0 \\ 0.0 \end{bmatrix}) &= 0.4 \\ p(\mathbf{x}_1|\mathbf{z}_{1} = \begin{bmatrix} 0.0 \\ 1.0 \end{bmatrix}) &= 0.6 \end{align}
Please answer the following questions:
(i) (6 points)
Solve for all four values of \hat{\alpha}(\mathbf{z}_t) and the value of c_t.
(ii) (4 points)
Solve for all four values of \hat{\beta}(\mathbf{z}_t).
(iii) (2 points)
What is the value of p(z_{0,1}|\mathbf{x}_1)?
Problem 2 (16 points)
Note: You’ll need a calculator for some things here, but you won’t on the exam.
You are running a particle filter with N=4 particles to track a 1D state z_t. At time t, your particles (which just underwent resampling) are:
z_{t,1} = 0 \quad z_{t,2} = 2 \quad z_{t,3} = 4 \quad z_{t,4} = 6
These particles all have equal weights, w_{t,i} = 0.25.
Step 1: Propagate. The particles are propagated through the state transition model z_{t+1} = z_t + \mathcal{N}(0, 1.0). For the sake of this problem, the random noise samples drawn for each particle are:
\epsilon_1 = 0.0 \quad \epsilon_2 = 0.2 \quad \epsilon_3 = -0.2 \quad \epsilon_4 = 0.1
(i) (3 points)
Calculate the locations of the new particles z_{t+1, i} after propagation.
Step 2: Weight. A new observation x_{t+1} = 2.0 is received. The observation likelihood is p(x | z) \propto \exp\left(-\frac{1}{2}(x - z)^2\right) (i.e., \sigma_{\text{obs}}^2 = 1.0). You may use the following approximations:
e^{-2.0} \approx 0.135 \quad e^{-0.02} \approx 0.980 \quad e^{-1.62} \approx 0.198 \quad e^{-8.405} \approx 0.0002
(ii) (6 points)
Calculate the new unnormalized weights w'_{t+1, i} for your four propagated particles. (You can ignore the constant 1/\sqrt{2\pi} as it will cancel.)
(iii) (4 points)
Calculate the normalized weights w_{t+1, i}.
(iv) (3 points)
Observe the distribution of the normalized weights you calculated in (iii). What problem does this demonstrate? In one or two sentences, describe this problem and its consequence for the particle filter.
Problem 3 (8 points)
Show that the posterior variance is independent of the observed values \mathbf{y}, i.e., argue that \text{Cov}[f_*|\mathbf{X}, \mathbf{y}, \mathbf{x}_*] does not depend on \mathbf{y}, only on \mathbf{X} and \mathbf{x}_*. Interpret what this means in the context of Gaussian Process regression. Hint: Start by writing down the equation for the posterior covariance (Eq. 9.25) and inspect its terms.
Problem 4 (8 points)
Show that the composition k(\mathbf{x}, \mathbf{x}') = k_2(k_1(\mathbf{x}, \mathbf{x}')) where k_1 is a valid kernel and k_2: \mathbb{R} \to \mathbb{R} with non-negative Taylor coefficients, produces a valid kernel. Hint: Write the Taylor series for k_2(z) = \sum_{n=0}^\infty a_n z^n. Then use the closure properties of kernels from lecture.