  We introduce a family of implicit probabilistic integrators for initial value
problems (IVPs), taking as a starting point the multistep Adams-Moulton method.
The implicit construction allows for dynamic feedback from the forthcoming
time-step, in contrast to previous probabilistic integrators, all of which are
based on explicit methods. We begin with a concise survey of the
rapidly-expanding field of probabilistic ODE solvers. We then introduce our
method, which builds on and adapts the work of Conrad et al. (2016) and Teymur
et al. (2016), and provide a rigorous proof of its well-definedness and
convergence. We discuss the problem of the calibration of such integrators and
suggest one approach. We give an illustrative example highlighting the effect
of the use of probabilistic integrators - including our new method - in the
setting of parameter inference within an inverse problem.
