#minimal-i-map

Question 1: Single choice

hardAI-checkedProbabilistic & Graphical Models

For a positive pmf over $x_1, \ldots, x_4, y_1, \ldots, y_4$ you only know the blankets of the $x$ 's: $\mathrm{MB}(x_1) = \{x_2, y_1\}$ , $\mathrm{MB}(x_2) = \{x_1, x_3, y_2\}$ , $\mathrm{MB}(x_3) = \{x_2, x_4, y_3\}$ and $\mathrm{MB}(x_4) = \{x_3, y_4\}$ . Without adding independencies beyond those implied by these blankets, how does the pmf factorise?

$\frac{1}{Z}\prod_{i=1}^{3} m_i(x_i, x_{i+1}) \prod_{i=1}^{4} g_i(x_i, y_i)$ 1
$\frac{1}{Z}\, g(y_1, \ldots, y_4) \prod_{i=1}^{3} m_i(x_i, x_{i+1}) \prod_{i=1}^{4} g_i(x_i, y_i)$ 2
$\frac{1}{Z}\prod_{i=1}^{3} m_i(x_i, x_{i+1}) \prod_{i=1}^{4} g_i(x_i, y_i) \prod_{i=1}^{3} h_i(y_i, y_{i+1})$ 3
$\frac{1}{Z}\, g(x_1, \ldots, x_4) \prod_{i=1}^{4} g_i(x_i, y_i)$ 4

Timed test