Many important real-world decision-making problems involve interactions of individuals with purely informational externalities, for example, in jury deliberations, expert committees, etc. We model such interactions of rational agents in a group, where they receive private information and act based on that information while also observing other people’s beliefs or actions. As a Bayesian agent attempts to infer the truth from her sequence of observations of actions of others and her own private signal, she recursively refines her belief on the signals that other players could have observed and actions that they could have taken given that other players are also rational. The existing literature addresses asymptotic properties of Bayesian group decisions (important questions such as convergence to consensus and learning). In this work, we address the computations that the Bayesian agent should undertake to realize the optimal actions at every decision epoch. We use the iterated eliminations of infeasible signals (IEIS) to model the thinking process as well as the calculations of a Bayesian agent in a group decision scenario. We show that IEIS algorithm runs in exponential time; however, when the group structure is a partially ordered set the Bayesian calculations simplify and polynomial-time computation of the Bayesian recommendations is possible. We next shift attention to the case where agents reveal their beliefs (instead of actions) at every decision epoch. We analyze the computational complexity of the Bayesian belief formation in groups and show that it is NP-hard. We also investigate the factors underlying this computational complexity and show how belief calculations simplify in special network structures or cases with strong inherent symmetries. We finally give insights about the statistical efficiency (optimality) of the beliefs and its relations to computational efficiency. Bayesian Group Decisions: Algorithms and Complexity