A wide range of applied problems may be reduced to an under-determined inverse problem of the form: find the "best" density f subject to a set of constraints. The problem of choosing the best distribution to represent the data elicited from an expert is one such example. Expert elicitation is a common practice in a range of applications where data regarding a quantity of interest is unavailable or uncertain. To solve for an optimal density, one of a range of information-theoretic, or norm related, measures is introduced over the feasible set. A particular difficulty arises when some constraints cannot be expressed as moments. Such is the case when the density is known to be unimodal, and its unique mode is given. In this thesis, we formulate and solve the problem of unimodal density estimation when using the minimum cross entropy measure. In particular, using several previous results by Khinchin, Shapp, and Kempermann we formulate the general nonlinear optimisation problem, and then solve it in two steps. The first step studies a relaxed form of the general problem. Necessary and sufficient conditions under which solutions of the relaxed problem results in unimodal densities are derived. In the second step, the general optimal solutions are obtained, by establishing convexity properties, and using duality and variational calculus methods. The problem of choosing the best distribution to represent the data elicited from an expert is formulated as an inverse problem using the minimum cross entropy measure. Throughout the thesis, the analytical results for unimodal density estimation are illustrated in the context of expert elicitation using the 4-step elicitation protocol. This protocol elicits information corresponding to the mode and an inter-percentile of the distribution representing the expert's opinion. Finally, the general problem of decision making under conditions of uncertainty is examined. A method is proposed, combining previously suggested safety first an