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Randomized algorithms are used in many state-of-the-art solvers for constraint satisfaction problems (CSP) and Boolean satisfiability (SAT) problems. For many of these problems, there is no single solver which will dominate others. Having access to the underlying runtime distributions (RTD) of these solvers can allow for better use of algorithm selection, algorithm portfolios, and restart strategies. Previous state-of-the-art methods directly try to predict a fixed parametric distribution that the input instance follows. In this paper, we extend RTD prediction models into the Bayesian setting for the first time. This new model achieves robust predictive performance in the low observation setting, as well as handling censored observations. This technique also allows for richer representations which cannot be achieved by the classical models which restrict their output representations. Our model outperforms the previous state-of-the-art model in settings in which data is scarce, and can make use of censored data such as lower bound time estimates, where that type of data would otherwise be discarded. It can also quantify its uncertainty in its predictions, allowing for algorithm portfolio models to make better informed decisions about which algorithm to run on a particular instance.