6 Hybrid Domains

Up until now, we have assumed that all variables and features are discrete; however, most real-world applications also contain continuous ones. Hybrid Markov Logic Networks (HMLNs) have been introduced in [14] and implemented in Alchemy. In this framework, continuous variables can appear as features.

Inference is performed by hybrid versions of MaxWalkSat and MC-SAT and
discriminative weight learning has been extended to handle hybrid clauses. For
the user, the only change is the allowed syntax of formulas. Currently, we are
able to handle formulas of the form `w B(X) * P(X)`, where `w` is a
weight (empty if performing weight learning, `B(X)` is a Boolean clause of
some variable set `X` (an indicator variable) and `P(X)` is an arbitrary
polynomial of `X`.

In a future release, arbitrary numeric terms will be allowed and Alchemy will
multiply it out to a sum of clauses of the form above. Also, inference and
learning algorithms will be extended to handle non-polynomials. In this case,
the user must supply the first and second derivatives of the formulas.

Marc Sumner 2010-01-22