Gibbs-Sampling

Method of approximation for variable elimination implemented in python

$$ \begin{align} P(A | B = b^{1}) &= \frac{1}{Z}\sum_{DC}(\Phi_1(A,B) \Phi_2(B,C) \Phi_3(C, D) \Phi_4(D,A))_{B = b^1} (Obs\,B = b^{1})\\ &= \frac{1}{Z}\sum_{DC}((\Phi_1(A) \Phi_2(C) \Phi_3(C, D) \Phi_4(D,A))_{B = b^1}(Elimination\, Order\, = C,D)\\ &= \frac{1}{Z}(\Phi_1(A) \sum_{D} \Phi_4(D,A)\sum_{C}\Phi_3(C,D) \Phi_2(C))(Sum\, out\, C)\\ &= \frac{1}{Z}(\Phi_1(A) \sum_{D} \Phi_4(D,A)\Phi_5(D))(Sum\,out\, D)\\ &= \frac{1}{Z}(\Phi_1(A) \Phi_6(A))\\ &\approx \end{align} $$

We show the process of the resulting factor tables:

Gibbs sampling procedure:

Randomly choosing A,C or D then sampling based on the posterior distribution of their Markov blanket. The convergence of the sampling procedure is shown in the figure below. The red line represents the value gained from variable elimination for $P(A = a1|B = b1)$