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The purpose of this thesis is to develop Bayesian methodology together with the proper computational tools to address two different problems. The first problem which is more general from a methodological point of view appears in computer experiments. We consider emulation of realizations of a monotone function at a finite set of inputs available from a computationally intensive simulator. We develop a Bayesian method for incorporating the monotonicity information in Gaussian process models that are traditionally used as emulators. The resulting posterior in the monotone emulation setting is difficult to sample from due to the restrictions caused by the monotonicity constraint. To overcome the difficulties faced in sampling from the constrained posterior was the motivation for development of a variant of sequential Monte Carlo samplers that are introduced in the beginning of this thesis. Our proposed algorithm that can be used in a variety of frameworks is based on imposition of the constraint in a sequential manner. We demonstrate the applicability of the sampler to different cases by two examples; one in inference for differential equation models and the second in approximate Bayesian computation. The second focus of the thesis is on an application in the area of particle physics. The statistical procedures used in the search for a new particle are investigated and a Bayesian alternative method is proposed that can address decision making and inference for a class of problems in this area. The sampling algorithm and components of the model used for this application are related to methods used in the first part of the thesis.