  When collaborating with an AI system, we need to assess when to trust its
recommendations. If we mistakenly trust it in regions where it is likely to
err, catastrophic failures may occur, hence the need for Bayesian approaches
for probabilistic reasoning in order to determine the confidence (or epistemic
uncertainty) in the probabilities in light of the training data. We propose an
approach to overcome the independence assumption behind most of the approaches
dealing with a large class of probabilistic reasoning that includes Bayesian
networks as well as several instances of probabilistic logic. We provide an
algorithm for Bayesian learning from sparse, albeit complete, observations, and
for deriving inferences and their confidences keeping track of the dependencies
between variables when they are manipulated within the unifying computational
formalism provided by probabilistic circuits. Each leaf of such circuits is
labelled with a beta-distributed random variable that provides us with an
elegant framework for representing uncertain probabilities. We achieve better
estimation of epistemic uncertainty than state-of-the-art approaches, including
highly engineered ones, while being able to handle general circuits and with
just a modest increase in the computational effort compared to using point
probabilities.
