Bayesian Inference: Observations and Applications discusses standard Bayesian inference, in which a-priori distributions are standard probability distributions. In some cases, however, a more general form of a-priori distributions (fuzzy a-priori densities) is suitable to model a-priori information. The combination of fuzziness and stochastic uncertainty calls for a generalization of Bayesian inference, i.e. fuzzy Bayesian inference. The authors explain how Bayes theorem may be generalized to handle this situation. Next, they present a decision analytic framework for completing selection of optimal parameters for machining process definition. In addition, a discussion section on the subjects of inference, experimental design, and risk aversion is included. The concluding review focuses on the sparse Bayesian methods from their model specifications, interference algorithms, and applications in sensor array signal processing. Sparse and structured sparse Bayesian methods formulate problems in a probabilistic manner by constructing a hierarchical model, allowing for the obtainment of flexible modeling capability and statistical information. (Bayesian Inference: Observations and Applications discusses standard Bayesian inference, in which a-priori distributions are standard probability distributions. In some cases, however, a more general form of a-priori distributions (fuzzy a-priori densities) is suitable to model a-priori information. The combination of fuzziness and stochastic uncertainty calls for a generalization of Bayesian inference, i.e. fuzzy Bayesian inference. The authors explain how Bayes theorem may be generalized to handle this situation. Next, they present a decision analytic framework for completing selection of optimal parameters for machining process definition. In addition, a discussion section on the subjects of inference, experimental design, and risk aversion is included. The concluding review focuses on the sparse Bayesian methods from their model specifications, interference algorithms, and applications in sensor array signal processing. Sparse and structured sparse Bayesian methods formulate problems in a probabilistic manner by constructing a hierarchical model, allowing for the obtainment of flexible modeling capability and statistical information.