Towards Integrative Machine Learning and Knowledge Extraction by Andreas Holzinger Randy Goebel Massimo Ferri & Vasile Palade

Author:Andreas Holzinger, Randy Goebel, Massimo Ferri & Vasile Palade
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

X takes terms while Y takes real numbers as values. The clause states that, for each X such that object(X) is true, the values of Y such that g(X,Y) is true, follow a Gaussian distribution with mean 0 and variance 1. You can think of an atom such as g(a,Y) as an encoding of a continuous random variable associated to term g(a). In DC you can express the same density as

where := indicates implication and continuous random variables are represented as terms that denote a value from a continuous domain. It is possible to translate DC into programs for cplint and in fact cplint allows also the DC syntax, automatically translating DC into its own syntax.

A semantics for hybrid programs was given independently in [16, 20, 28]. In [28] the semantics of Hybrid Probabilistic Logic Programs (HPLP) is defined by means of a stochastic generalization STp of the Tp operator that applies the sampling interpretation of the distribution semantics to continuous variables: STp is applied to interpretations that contain ground atoms (as in standard logic programming) and terms of the form where t is a term indicating a continuous random variable and v is a real number. If the body of a clause is true in an interpretation I, STp(I) will contain a sample from the head.

The authors of [20] define a probability space for N continuous random variables by considering the Borel -algebra over and fixing a Lebesgue measure on this set as the probability measure. The probability space is lifted to cover the entire program using the least model semantics of constraint logic programs.

If an atom encodes a continuous random variable (such as g(X,Y) above), asking for the probability that a ground instantiation, such as g(a,0.3), is true is not meaningful, as the probability that a continuous random variables takes a specific value is always 0. In this case you want to compute the probability that the random variable falls in an interval or you want to know its density, possibly after having observed some evidence. If the evidence is on an atom defining another continuous random variable, the definition of conditional probability cannot be applied, as the probability of the evidence would be 0 and so the fraction would be undefined. This problem is tackled in [28] by providing a definition using limits.

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