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This allows for the underlying inference to be solved quickly as a convex optimization problem.
This is useful in problems such as [[collective classification]], [[link prediction]], [[social network]] modelling, and [[record linkage|object identification/entity resolution/record linkage]].
== Semantics ==
=== HL-MRF ===
A PSL program defines a family of probabilistic [[graphical model | probabilistic graphical models]] that are parameterized by data.
More specifically, the family of graphical models it defines belongs to a special class of [[Markov random field]] known as a Hinge-Loss Markov Field (HL-MRF).
A HL-MRF determines a density function over a set of continuous variables <math>\mathbf{y} = (y_1, \cdots , y_n)</math> with joint ___domain <math>[0, 1]^n</math> using set of evidence <math>\mathbf{x} = (x_1, \cdots , x_m)</math>, weights <math>\mathbf{w} = (w_1, \cdots , w_m)</math>, and potential functions <math>\mathbf{\phi} = (\phi_1, \cdots , \phi_k)</math> of the form <math>\mathbf{\phi_i (\mathbf{x}, \mathbf{y})} = \max(\ell_i (\mathbf{x}, \mathbf{y}), 0)^{d_i}</math> where <math>\ell_i</math> is a linear function and <math>d_i \in \{1,2\}</math>.
The conditional distribution of <math>\mathbf{y}</math> given the observed data <math>\mathbf{x}</math> is defined as: <br>
<math>
P(\mathbf{y} | \mathbf{x}) = \frac{1}{Z(\mathbf{y})} \exp(\sum_{i = 1}^{k} w_i \phi_i (\mathbf{x}, \mathbf{y}))
</math>
This density is a [[Logarithmically convex function]], and thus finding a [[Maximum a posteriori estimation]] of the joint state of <math>\mathbf{y}</math> is a convex problem.
This useful property allows inference in PSL to be solvable in polynomial-time.
== See also ==
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