In this paper, we present a Bayesian approach to estimate the mean of a binary variable and changes in the mean over time, when the variable is subject to misclassification error. These parameters are partially identified, and we derive identified sets under various assumptions about the misclassification rates. We apply our method to estimating the prevalence and trend of prescription opioid misuse, using data from the 2002-2014 National Survey on Drug Use and Health. Using a range of priors, the posterior distribution provides evidence that among middle-aged White men, the prevalence of opioid misuse increased multiple times between 2002 and 2012.