In spite of the rapidly increasing number of applications of machine learning in various domains, a principled and systematic approach to the incorporation of domain knowledge in the engineering process is still lacking and ad hoc solutions that are difficult to validate are still the norm in practice, which is of growing concern not only in mission-critical applications. In this note, we introduce Probabilistic Approximate Logic (PALO) as a logic based on the notion of mean approximate probability to overcome conceptual and computational difficulties inherent to strictly probabilistic logics. The logic is approximate in several dimensions. Logical independence assumptions are used to obtain approximate probabilities, but by averaging over many instances of formulas a useful estimate of mean probability with known confidence can usually be obtained. To enable efficient computational inference, the logic has a continuous semantics that reflects only a subset of the structural properties of classical logic, but this imprecision can be partly compensated by richer theories obtained by classical inference or other means. Computational inference, which refers to the construction of models and validation of logical properties, is based on Stochastic Gradient Descent (SGD) and Markov Chain Monte Carlo (MCMC) techniques and hence another dimension where approximations are involved. We also present the Logical Imagination Engine (LIME), a prototypical implementation of PALO based on TensorFlow. Albeit not limited to the biological domain, we illustrate its operation in a quite substantial bioinformatics machine learning application concerned with network synthesis and analysis in a recent DARPA project.
@cite_14 is another approach to overcome the fact that a probabilistic interpretation of formulas is not truth-functional by using a less abstract semantics that interprets each formula as the set of assignments for which it holds so that conjunction becomes a simple intersection. Although this is an elegant solution, with our mean probability semantics that includes lower and upper bounds, it turns out that the bounds are sufficiently tight so that replacing our approximate by a strict probabilistic interpretation is unnecessary for the data-rich applications we are targeting. Two other practical difficulties with an exact probabilistic semantics are that dependencies between subformulas referring to external data are often unknown and even if all known dependencies would be taken into account it would lead to an unacceptably high computational complexity in the context of model generation and learning.
@cite_14
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Mechanisms for the automation of uncertainty are required for expert systems. Sometimes these mechanisms need to obey the properties of probabilistic reasoning. We argue that a purely numeric mechanism, like those proposed so far, cannot provide a probabilistic logic with truth functional connectives. We propose an alternative mechanism, Incidence Calculus, which is based on a representation of uncertainty using sets of points, which might represent situations models or possible worlds. Incidence Calculus does provide a probabilistic logic with truth functional connectives.