Keywords: Machine learning, machine reasoning, AI, statistical machine learning, computational methods
We briefly review and give references for some paradigms that define integrated models of statistical and logical inference. Our main perspective is that the fields of learning and reasoning are jointly claimed by many disciplines; if we would like to use a characterization such as “deep” in this area, then as part of doing that we should give appreciation to some of the underlying classical perspectives involved. In this context, we make note of (1.) paradigms for reasoning about data in space and time in physics using a variety of methods such as tensors, (2.) paradigms that are now emerging that are giving the relationships among multi-layer neural networks, hierarchical Gaussian processes, and kernel methods, and that at some point may be able to relate those to multi-level models in which the different levels may be of different algebraic types, (3.) a full set of perspectives on types by B. Jacobs and others in which different types can come with different kinds of rules of deductive or probabilistic inference, (4.) perspectives from modern logic where a logic is a formal language over a set of types and where the concepts of “and” and “or” can relate to different concepts of “*” and “+”, and (5.) paradigms outside the hard sciences such as the van Hiele model in education, the five levels of understanding of which can apply to levels of understanding by machines. For those who would like to explore topics further we provide a handout with an annotated bibliography on the specific methods we discuss.