On estimation of probability of events from imperfect Bernoulli trials

In many experiments, Bernoulli trials are conducted to estimate the probability of an event of interest. The outcomes of the trials are usually known without error, that is, we know with certainty that an event occurred or not. Estimation of the probability of the event then proceeds along lines that can be found in standard textbooks on probability. The problem gets a bit more complicated if the data obtained in a Bernoulli experiment carry uncertainty about the event of interest. We call such trials imperfect Bernoulli trials as opposed to perfect trials when the outcomes of the experiment are known without error. The probability estimation in the case of imperfect trials must be modified to take into account the uncertainties. A complete Bayesian procedure is developed for this purpose. It provides an update formula for the posterior density of the probability of interest as data from new trials are obtained. The work on this problem has been motivated by studies in neurophisiology where large sets of patch-clamp recordings of synaptic currents are processed to estimate the probability of a synaptic event. As example, we present application of the methodology to simulated synaptic currents.