Inferences with an Unknown Noise Level in a Bernoulli Process

Inferences about a proportion p are often based on data generated from dichotomous processes, which are generally modeled as processes that are Bernoulli in p. In reality, the assumption that a data-generating process is Bernoulli in p is often violated due to the presence of noise. The level of noise is usually unknown and, furthermore, dependent on the unknown proportion in which one is interested. A specific model which takes into account the existence of noise is developed. Any arguments about p based exclusively on a likelihood analysis can lead to difficulties. A Bayesian approach is used, which also helps us to formalize a priori dependence between the proportion and the noise level. Empirical data are used to illustrate the model and provide some flavor of the implications of our uncertainty about the noise for inferences about a proportion.