Since the data x is fixed, the posterior is proportional to the joint distribution over all of the variables and data: Using the chain rule, for some i we can then factor the joint distribution as: Since x is fixed, we can also write While updating the parameter θ i , θ ¬i is held fixed, which means that the second term p(θ ¬i |x) is constant and the factored joint distribution is proportional to the complete conditional, thus: Since these two expressions are proportional, setting θ i to the mode of its complete conditional will only increase the value of the posterior probability.
our update rule for the variable θ i is given as: Since this update equation monotonically increases the estimate of the posterior probability, and the value of the posterior probability is bounded above by 1, we can use the monotone convergence theorem to conclude that this algorithm will converge to a local maximum in the posterior distribution.