Variable elimination

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Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields.[1][2] It can be used for inference of maximum a posteriori (MAP) state or estimation of marginal distribution over a subset of variables. The algorithm has exponential time complexity, but could be efficient in practice for the low-treewidth graphs, if the proper elimination order is used.


The most common query type is in the form where and are disjoint subsets of , and is observed taking value . A basic algorithm to computing p(X|E = e) is called variable elimination (VE), first put forth in.[2]
Algorithm 1, called sum-out (SO), eliminates a single variable from a set of potentials,[3] and returns the resulting set of potentials. The algorithm collect-relevant simply returns those potentials in involving variable .

Algorithm 1 sum-out(,)

= collect-relevant(,)
= the product of all potentials in


Algorithm 2, taken from,[2] computes from a discrete Bayesian network B. VE calls SO to eliminate variables one by one. More specifically, in Algorithm 2, is the set C of CPTs for B, is a list of query variables, is a list of observed variables, is the corresponding list of observed values, and is an elimination ordering for variables , where denotes .

Algorithm 2 VE()

Multiply evidence potentials with appropriate CPTs While σ is not empty
Remove the first variable from
= sum-out
= the product of all potentials



  1. Zhang, N.L., Poole, D.: A Simple Approach to Bayesian Network Computations. In:7th Canadian Conference on Artificial Intelligence, pp. 171–178. Springer, New York(1994)
  2. 2.0 2.1 2.2 Zhang, N.L., Poole, D.:A Simple Approach to Bayesian Network Computations.In: 7th Canadian Conference on Artificial Intelligence,pp. 171--178. Springer, New York (1994)
  3. Koller,D.,Friedman,N.:ProbabilisticGraphicalModels:PrinciplesandTechniques. MIT Press, Cambridge, MA (2009)