#change-of-measure

Question 1: Single choice

hardAI-checkedProbabilistic & Graphical Models

An HMM posterior has the form $p(\mathbf{x} \mid y_{1:t}) \propto \underbrace{f_1(x_1)\prod_{i=2}^t f_i(x_i \mid x_{i-1})}_{f(\mathbf{x})}\;\prod_{i=1}^t g_i(x_i),$ where $g_i(x_i) = \phi_i^y(x_i, y_i)$ for the observed $y_i$ . You draw $n$ trajectories $\mathbf{x}^{(k)} \sim f$ by ancestral sampling and approximate $\mathbb{E}[h(\mathbf{x}) \mid y_{1:t}] \approx \sum_k W^{(k)} h(\mathbf{x}^{(k)})$ . Which weights are correct?

$W^{(k)} \propto \prod_{i=1}^t g_i(x_i^{(k)})$ , normalised to sum to one1
$W^{(k)} = 1/n$ 2
$W^{(k)} \propto f(\mathbf{x}^{(k)})$ , normalised to sum to one3
$W^{(k)} \propto f(\mathbf{x}^{(k)}) \prod_{i=1}^t g_i(x_i^{(k)})$ , normalised to sum to one4

Timed test