We consider the problem of statistical inference of binomial proportions for non-matched, correlated samples, under the Bayesian framework. Such inference can arise when the same group is observed a different number of times on two or more inference occasions, with the aim of testing the proportion of some trait. These scenarios can occur when we are interested to infer the proportion of extreme wave height per year, at a certain measuring station, where measurements are made every hour. Gaps in measurements, either due to a malfunction of the measuring instrument or another reason, can result in an unequal number of observations in different years. For such scenarios, we develop an adaptive Bayesian method, and suggest a heuristic decision procedure to conduct statistical inference. We use the ø-divergence measure to quantify the perturbation of the posterior distribution of the proportion in different time points. We present a simulation study for frequentist power investigation for both the adaptive Bayesian method, as well as the regular frequentist method, using the Monte Carlo technique. Based on the simulation study of frequentist power, as well as theoretical proof, under certain design, the adaptive Bayesian method is shown to outperform the regular frequentist method. We administer the developed adaptive Bayesian method to two case studies when the total number of observation instances of the same group are unequal, at different time points of interest.