#forward-backward

Question 1: Single choice

hardAI-checkedProbabilistic & Graphical Models

An HMM has transitions $a_{p,q}$ and emissions $b_q(o)$ . Its forward and backward terms are $\alpha_j(p) = P(x_1, \ldots, x_j, y_j = p)$ and $\beta_j(p) = P(x_{j+1}, \ldots, x_n \mid y_j = p)$ . Which expression equals $P(y_2 = 1, y_3 = 2, y_4 = 1 \mid x)$ ?

$\alpha_2(1)\,a_{1,2}\,b_2(x_3)\,a_{2,1}\,b_1(x_4)\,\beta_4(1) \,/\, P(x)$ 1
$\alpha_2(1)\,a_{1,2}\,b_2(x_3)\,a_{2,1}\,b_1(x_4)\,\beta_4(1)$ 2
$\alpha_2(1)\,a_{1,2}\,a_{2,1}\,\beta_4(1) \,/\, P(x)$ 3
$\alpha_2(1)\,\alpha_3(2)\,\alpha_4(1) \,/\, P(x)$ 4

Timed test