We provide behavioral characterizations of notions of stochastic independence between two sources of uncertainty for agents with ambiguity-sensitive preferences. These notions, inspired from the literature on imprecise probabilities, include the independent product of two sets of marginal beliefs proposed by Gilboa and Schmeidler (J Math Econ 18(2): 141–153, 1989) in their seminal paper. Our analysis draws a clear relationship between the property of dynamic consistency of preferences and some form of stochastic independence between the relevant events.