Probabilistic programming languages are often distinguished by their support for higher-order functions. We demonstrate that a more nuanced distinction is possible if the focus is set on whether the program induces stochastic control flow in recursions used by higher-order functions. In the case where higher-order functionality is only applied to deterministically known values we can compile a higher-order program into a probabilistic graphical model without restricting expressivity. We provide the implementation of this approach in our probabilistic programming compiler that previously only supported first-order functional abstractions.