Variable elimination

Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields. 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 ${\displaystyle f}$, also known as a potential, of variables ${\displaystyle V}$ is a relation between each instantiation of ${\displaystyle v}$ of variables ${\displaystyle f}$ to a non-negative number, commonly denoted as ${\displaystyle f(x)}$. 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. 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.

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