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In spite of the rapidly increasing number of applications of machine learningin various domains, a principled and systematic approach to the incorporationof domain knowledge in the engineering process is still lacking and ad hocsolutions that are difficult to validate are still the norm in practice, whichis of growing concern not only in mission-critical applications.  In this note, we introduce Probabilistic Approximate Logic (PALO) as a logicbased on the notion of mean approximate probability to overcome conceptual andcomputational difficulties inherent to strictly probabilistic logics. The logicis approximate in several dimensions. Logical independence assumptions are usedto obtain approximate probabilities, but by averaging over many instances offormulas a useful estimate of mean probability with known confidence canusually be obtained. To enable efficient computational inference, the logic hasa continuous semantics that reflects only a subset of the structural propertiesof classical logic, but this imprecision can be partly compensated by richertheories obtained by classical inference or other means. Computationalinference, which refers to the construction of models and validation of logicalproperties, is based on Stochastic Gradient Descent (SGD) and Markov ChainMonte Carlo (MCMC) techniques and hence another dimension where approximationsare involved.  We also present the Logical Imagination Engine (LIME), a prototypicalimplementation of PALO based on TensorFlow. Albeit not limited to thebiological domain, we illustrate its operation in a quite substantialbioinformatics machine learning application concerned with network synthesisand analysis in a recent DARPA project.

Probabilistic Approximate Logic and its Implementation in the Logical Imagination Engine