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In this thesis, I investigate the question “How can we discover entities in the world that we cannot observe directly?” I construe “entity” loosely, but pay special attention to theoretical entities—in contrast to objects that can be directly measured—in scientific theories, especially psychological “properties” studied by psychometricians. My research focuses on the apparent conflict between formal, statistical inference methods that have become more and more widespread in the social sciences, and claims by scientists such as Bartholomew and philosophers such as Hempel that such methods, as implemented in digital computers, cannot uncover important scientific theoretical entities. I defend the spirit, though not the specifics, of past statistical work in this area, by significantly improving upon older methods with new methods of my design, and I demonstrate with mathematical proofs, simulations, and applications to real data, that these methods can accomplish the tasks that were claimed to be impossible. Alongside that work, I also identify some of the limitations of such approaches, potential strategies for overcoming these limitations in the future, and lay the foundations for the development of more methods in very different theoretical and scientific domains.