In this chapter, we investigate the representation of conditional expectation values for partially observed diffusion processes in terms of appropriate estimators. The work of Kalman and Bucy has established a duality between filtering and estimation in the context of time-continuous linear systems. This duality has recently been extended to time-continuous nonlinear systems in terms of an optimization problem constrained by a backward stochastic partial differential equation. Here we revisit this problem from the perspective of appropriate forward-backward stochastic differential equations. Our approach sheds new light on the conditional estimation problem and provides a unifying perspective. It is also demonstrated that certain formulations of the estimation problem lead to deterministic formulations similar to the linear Gaussian case as originally investigated by Kalman and Bucy. Finally, we discuss an application of the proposed formulation to optimal control problem on partially observed diffusion processes.