Markov logic network

A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, defining probability distributions on possible domains of a given signature. Markov logic networks generalize first-order logic, in the sense that, in a certain limit, all unsatisfiable statements have a probability of zero, and all tautologies have probability one.

History

In 2002, Ben Taskar, Pieter Abbeel and Daphne Koller introduced relational Markov networks as templates to specify Markov networks abstractly and without reference to a specific domain. Work on Markov logic networks began in 2003 by Pedro Domingos and Matt Richardson.

Description

Briefly, it is a collection of formulas from first-order logic, to each of which is assigned a real number, the weight. Taken as a Markov network, the vertices of the network graph are atomic formulas, and the edges are the logical connectives used to construct the formula. Each formula is considered to be a clique, and the Markov blanket is the set of formulas in which a given atom appears. A potential function is associated to each formula, and takes the value of one when the formula is true, and zero when it is false. The potential function is combined with the weight to form the Gibbs measure and partition function for the Markov network.

The above definition glosses over a subtle point: atomic formulas do not have a truth value unless they are grounded and given an interpretation; that is, until they are ground atoms with a Herbrand interpretation. Thus, a Markov logic network becomes a Markov network only with respect to a specific grounding and interpretation; the resulting Markov network is called the ground Markov network. The vertices of the graph of the ground Markov network are the ground atoms. The size of the resulting Markov network thus depends strongly (exponentially) on the number of constants in the domain of discourse.

Semantics

Together with a given domain, a Markov logic network defines a probability distribution on the set of all interpretations of its predicates on the given domain. It does so by inducing a Markov network whose nodes are ground atomic formulas.

Inference

The goal of inference in a Markov logic network is to find the stationary distribution of the system, or one that is close to it; that this may be difficult or not always possible is illustrated by the richness of behaviour seen in the Ising model. As in a Markov network, the stationary distribution finds the most likely assignment of probabilities to the vertices of the graph; in this case, the vertices are the ground atoms of an interpretation. That is, the distribution indicates the probability of the truth or falsehood of each ground atom. Given the stationary distribution, one can then perform inference in the traditional statistical sense of conditional probability: obtain the probability $P(A|B)$ that formula A holds, given that formula B is true.

Inference in MLNs can be performed using standard Markov network inference techniques over the minimal subset of the relevant Markov network required for answering the query. These techniques include Gibbs sampling, which is effective but may be excessively slow for large networks, belief propagation, or approximation via pseudolikelihood.

See also

• Markov random field
• Statistical relational learning
• Probabilistic logic network
• Probabilistic soft logic

External links

• University of Washington Statistical Relational Learning group
• Alchemy 2.0: Markov logic networks in C++
• pracmln: Markov logic networks in Python
• ProbCog: Markov logic networks in Python and Java that can use its own inference engine or Alchemy's
• markov thebeast: Markov logic networks in Java
• RockIt: Markov logic networks in Java (with web interface/REST API)
• Tuffy: A Learning and Inference Engine with strong RDBMs-based optimization for scalability
• Felix: A successor to Tuffy, with prebuilt submodules to speed up common subtasks
• Factorie: Scala based probabilistic inference language, with prebuilt submodules for natural language processing etc
• Figaro: Scala based MLN language
• LoMRF: Logical Markov Random Fields, an open-source implementation of Markov Logic Networks in Scala