Variable elimination

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields.[1] 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.


Enabling a key reduction in algorithmic complexity, a factor , also known as a potential, of variables is a relation between each instantiation of of variables to a non-negative number, commonly denoted as .[2] A factor does not necessarily have a set interpretation. One may perform operations on factors of different representations such as a probability distribution or conditional distribution.[2] Joint distributions often become too large to handle as the complexity of this operation is exponential. Thus variable elimination becomes more feasible when computing factorized entities.

Basic Operations[edit]

Variable Summation[edit]

Algorithm 1, called sum-out (SO), or marginalization, eliminates a single variable from a set of factors,[3] and returns the resulting set of factors. The algorithm collect-relevant simply returns those factors in involving variable .

Algorithm 1 sum-out(,)

= collect factors relevant to
= the product of all factors in



Here we have a joint probability distribution. A variable, can be summed out between a set of instantiations where the set at minimum must agree over the remaining variables. The value of is irrelevant when it is the variable to be summed out. [2]

true true true false false 0.80
false true true false false 0.20

After eliminating , its reference is excluded and we are left with a distribution only over the remaining variables and the sum of each instantiation.

true true false false 1.0

The resulting distribution which follows the sum-out operation only helps to answer queries that do not mention .[2] Also worthy to note, the summing-out operation is commutative.

Factor Multiplication[edit]

Computing a product between multiple factors results in a factor compatible with a single instantiation in each factor.[2]

Algorithm 2 mult-factors(,)[2]

= Union of all variables between product of factors
= a factor over where for all
For each instantiation
For 1 to
instantiation of variables consistent with

Factor multiplication is not only commutative but also associative.


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.[1]

Taken from,[1] this algorithm 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 .

Variable Elimination Algorithm VE()

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



Finding the optimal order in which to eliminate variables is an NP-hard problem. As such there are heuristics one may follow to better optimize performance by order:

  1. Minimum Degree: Eliminate the variable which results in constructing the smallest factor possible.[2]
  2. Minimum Fill: By constructing an undirected graph showing variable relations expressed by all CPTs, eliminate the variable which would result in the least edges to be added post elimination.[2]


  1. ^ a b c 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. ^ a b c d e f g h Darwiche, Adnan (2009-01-01). Modeling and Reasoning with Bayesian Networks. doi:10.1017/cbo9780511811357. ISBN 9780511811357. 
  3. ^ Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge, MA (2009)