The notion of stable weak equivalence for actions on probability spaces has been introduced by Kechris and studied by many other authors including Abert, Bowen, Burton, Elek, Tucker-Drob, and Weiss. Particularly, it was shown by Bowen and Tucker-Drob that the space of stable weak equivalence classes of actions on the standard probability space of a fixed countable group is a metrizable simplex. In the amenable case, such a simplex coincides with the simplex of invariant random subgroups (IRS) of the group. In joint work with Burton, we initiated the study of stable weak equivalence for actions on the hyperfinite II1 factor and, more generally, tracial von Neumann algebras. In my talk I will provide an introduction to this subject, and present some of our new results.