Bridges between Deterministic and Probabilistic Models for Binary Data

For the analysis of binary data, various deterministic models have been proposed, which are generally simpler to fit and easier to understand than probabilistic models. We claim that corresponding to any deterministic model is an implicit stochastic model in which the deterministic model fits imperfectly, with errors occurring at random. In the context of binary data, we consider two error models in the first model, all predictions are equally likely to be in error; in the second model, the probability of error depends on the model prediction. We show how to fit these models using a stochastic modification of deterministic optimization schemes. The advantages of fitting the stochastic models explicitly (rather than implicitly, by simply fitting a deterministic model and accepting the occurrence of errors) include quantification of uncertainty in the deterministic model's parameter estimates, better estimation of the true model error rate, and the ability to check the fit of the model nontrivially. We illustrate with a simple theoretical example of item response data and with empirical examples from archaeology and the psychology of choice.