Bayesian prediction of functions with applications to manufacturing and marketing

Bayesian predictive methods have a number of advantages over traditional statistical methods. For one, Bayesian methods allow one to aggregate information from multiple sources (theoretical models, experimental data, and expert opinions). In addition, a Bayesian prediction can be updated dynamically as new data becomes available. Lastly, Bayesian methods are compatible with decision analytic methods such as value of information calculations. Hence, when using a Bayesian approach, there is no guesswork in optimal (profit maximizing) design of experiments. Bayesian methods do however have one primary drawback over traditional statistical methods and that is that they tend to be more mathematically complex. As a result, especially in complex problems, Bayesian methods see much less use than traditional statistical methods. One example of this is in the prediction of functions. From a mathematical perspective, a function is simply a list of numbers, or in other words a vector. Thus, when predicting a function, one is really just assigning a multivariate probability distribution. What makes this problem fundamentally difficult, however, is that functions are generally defined over a continuous space. Thus, when predicting a function, one must define an uncountably infinite dimensional probability distribution. Because of the high dimensionality, the general problem of Bayesian prediction of functions is very far from feasible. However, certain families of these probability distributions can be treated. In this work, functional probability distributions which satisfy a certain condition on their dependence structure (a chainlike dependence structure) will be considered. In the first chapter, Bayesian updating of these probability distributions will be discussed. In particular, we will show that the updated marginal distributions for any prediction which satisfies the condition on the dependence structure can be calculated numerically using particle filtered Markov chains. In addition, analytic solutions for the updated marginals will be given for two families of functional probability distributions. In the remaining chapters, these results will be applied to real world problems in manufacturing and marketing. In chapters 2 and 3, stability limit prediction in high speed machining will be considered while in chapter 4 the focus will be on demand curve prediction