A Bayesian comparison of different classes of dynamic models using empirical data

This paper deals with the Bayesian methods of comparing different types of dynamical structures for representing the given set of observations. Specifically, given that a given process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(·)</tex> obeys one of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> distinct stochastic or deterministic difference equations each involving a vector of unknown parameters, we compute the posterior probability that a set of observations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{y(1),...,y(N)}</tex> obeys the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> th equation, after making suitable assumptions about the prior probability distribution of the parameters in each equation. The difference equations can be nonlinear in the variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> but should be linear in the parameter vector in it. Once the posterior probability is known, we can find a decision rule to choose between the various structures so as to minimize the average value of a loss function. The optimum decision rule is asymptotically consistent and gives a quantitative explanation for the "principle of parsimony" often used in the construction of models from empirical data. The decision rule answers a wide variety of questions such as the advisability of a nonlinear transformation of data, the limitations of a model which yields a perfect fit to the data (i.e., zero residual variance), etc. The method can be used not only to compare different types of structures but also to determine a reliable estimate of spectral density of process. We compare the method in detail with the hypothesis testing method, and other methods and give a number of illustrative examples.