Answer Happy

Consider a Markov random field with pairwise potentials, P(y1, .
. . , y5) ∝ e P edge e ψe(ye) , over n variables, Y1, . . . , Yn,
that are binary-valued (yi ∈ {0, 1}) with the following potential
function: ψ(yi , yj ) = yi ∗ yj (i.e., 1 if both are 1 and 0
otherwise). What is P(y1 = y2 = y3 = y4 = 1) for each of the
following graph structures?
Consider a Markov random field with pairwise potentials, P(y₁,..., 5) x edge e (v), over n variables, Y₁,..., Yn, that are binary-valued (y; € {0,1}) with the following potential function: (y₁, y) = Yi * Yj (i.e., 1 if both are 1 and 0 otherwise). What is P(y₁ = y2 = y3 = y₁ = 1) for each of the following graph structures? For graduate students: What expression gives this probability (for each network) in general if there are n nodes instead of 4? (i) A chain network (20 points) Y₁ Y₂ Y₁ (ii) A star network (20 points) Y₂ Y Y₁ (iii) A complete network (20 points) Y₁ Y. Y Y₂