# Bayesian Non- and Semi-parametric Methods and Applications

Peter E. Rossi
Pages: 272
https://www.jstor.org/stable/j.ctt5hhrfp

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-xvi)
4. 1 Mixtures of Normals
(pp. 1-58)

In this chapter, I will review the mixture of normals model and discuss various methods for inference with special attention to Bayesian methods. The focus is entirely on the use of mixtures of normals to approximate possibly very high dimensional densities. Prior specification and prior sensitivity are important aspects of Bayesian inference and I will discuss how prior specification can be important in the mixture of normals model. Examples from univariate to high dimensional will be used to illustrate the flexibility of the mixture of normals model as well as the power of the Bayesian approach to inference for the...

5. 2 Dirichlet Process Prior and Density Estimation
(pp. 59-89)

One of the principal criticisms that can be leveled against the finite mixture of normals approach is that the approach is not non-parametric. A non-parametric approach would require the ability to accommodate a very large number of normal components and that as N approaches infinity the number of components could increase so as to flexibly fit any density. Although I have demonstrated that the finite mixture approach is feasible with a very large number of multivariate components (K> 10), a fixedKprocedure cannot make explicit non-parametric claims. Given that it is difficult to develop a sensible method of expanding...

6. 3 Non-parametric Regression
(pp. 90-114)

From a non-parametric point of view, a regression model is simply a model for the conditional distribution of the dependent variable,y, given the vector of independent variables,x. A full non-parametric approach should directly estimate or make inferences regarding this conditional distribution and, therefore, provide estimates of any functional of this distribution. Much of the large literature on non-parametric regression models focuses on the conditional mean function.

Various methods have been proposed to make the specification of the conditional mean function as flexible as possible and verify various non-parametric claims that as the sample size increases without bound the...

7. 4 Semi-parametric Approaches
(pp. 115-151)

As I have argued in the introduction and illustrated in Chapter 3, a fully non-parametric approach to Bayesian model inference will involve a non-trivial density estimation problem. For example, a truly non-parametric approach to regression involves all aspects of the conditional distribution ofy|xas the object of the modeling exercise. This makes great demands of the data, particularly for high dimensional problems. In many applications, there is insufficient data to apply a fully non-parametric approach. In these cases, we must rely on parametric methods for at least some parts of the problem. For example, single-index models in which a...

8. 5 Random Coefficient Models
(pp. 152-186)

Random coefficient models are frequently applied in a panel data setting in which the data consist of a set of cross-sectional units observed over time. For example, the Homescan data available from the Kilts Center of Marketing at the University of Chicago’s Booth School of Business tracks the purchases of some 30,000–60,000 households over a broad set of products. Each household is in the panel for a limited length of time, varying from less than one year to more than six years. In many applications, only a small and relatively homogenous set of products are considered. In these situations,...

9. 6 Conclusions and Directions for Future Research
(pp. 187-194)

The preceding chapters establish two important conclusions: (1) mixture of normal models with appropriately assessed informative priors provide a very useful density approximation method that is applicable in high dimensions without undue over-fitting; and (2) many important problems in micro-econometrics can be solved by using mixture of normals to approximate key densities in these models. The applications considered so far include non-parametric regression with a continuous dependent variable, semi-parametric regression with a single index model, semi-parametric inference for panel choice models with unknown distributions of heterogeneity, and semiparametric inference for instrumental variable models with linear structural and instrument equations.

Non-parametric...

10. Bibliography
(pp. 195-200)
11. Index
(pp. 201-202)