In many prediction tasks such as medical diagnostics, sequential decisions are crucial toprovide optimal individual treatment. Budget in real-life applications is always limited,and it can represent any limited resource suchas time, money, or side e↵ects of medications.In this contribution, we develop a POMDPbased framework to learn cost-sensitive heterogeneous cascading systems. We provideboth the theoretical support for the introduced approach and the intuition behind it.We evaluate our novel method on some standard benchmarks, and we discuss how thelearned models can be interpreted by humanexperts.