#posterior-sampling

Question 1: Single choice

hardAI-checkedProbabilistic & Graphical Models

Forward filtering backward sampling draws whole state trajectories from $p(h_{1:n} \mid v_{1:n})$ : sample $h_n \propto \alpha(h_n)$ , then go backwards with $p(h_{t-1} \mid h_t, v_{1:n})$ . Here $\alpha(h_t) = p(h_t, v_{1:t})$ and $\beta(h_t) = p(v_{t+1:n} \mid h_t)$ . Which quantities does the backward step $p(h_{t-1} \mid h_t, v_{1:n})$ need?

$\alpha(h_{t-1})$ , $\alpha(h_t)$ , $p(h_t \mid h_{t-1})$ and $p(v_t \mid h_t)$ 1
$\alpha(h_{t-1})$ , $\beta(h_t)$ , the transition $p(h_t \mid h_{t-1})$ and the emission $p(v_t \mid h_t)$ 2
Only the filtering distributions $p(h_{t-1} \mid v_{1:t-1})$ for each $t$ 3
$\beta(h_{t-1})$ , $\beta(h_t)$ , the transition $p(h_t \mid h_{t-1})$ and the emission $p(v_{t-1} \mid h_{t-1})$ 4

Timed test