Bayesian Econometrics: Conjugate Analysis and Rejection Sampling
In real-world problems we are invariably faced with making decisions in an environment of uncertainty (see also the chapter by R. Korsan in this volume). A statistical paradigm then becomes essential for extracting information from observed data and using this to improve our knowledge about the world (inference), and thus guiding us in the decision problem at hand. The underlying probability interpretation for a Bayesian is a subjective one, referring to a personal degree of belief. The rules of probability calculus are used to examine how prior beliefs are transformed to posterior beliefs by incorporating data information. The sampling model is a “window” [see Poirier (1988)] through which the researcher views the world. Here we only consider cases where such a model is parameterized by a parameter vector θ of finite dimension. A Bayesian then focuses on the inference on θ (treated as a random variable) given the observed data Y (fixed), summarized in the posterior density p(θ|Y). The observations in Y define a mapping from the prior p(θ) into p(θ|Y). This posterior distribution can also be used to integrate out the parameters when we are interested in forecasting future values, say, Ỹ, leading to the post-sample predictive density p(Ỹ|Y) = ∫ p(Ỹ|Y, θ)p(θ|Y)d θ where p(Ỹ|Y, θ) is obtained from the sampling model.
KeywordsPosterior Distribution Posterior Density High Posterior Density Prior Density High Posterior Density
Unable to display preview. Download preview PDF.
- Drèze J. H. (1962). “The Bayesian Approach to Simultaneous Equation Estimation,” O.N.R. Research Memorandum no. 67, Northwestern University.Google Scholar
- Good, I. J. (1976). “The Bayesian Influence, or How to Sweep Subjectivism Under the Carpet,” in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, vol. II, W. Harper and C. Hooker (eds.), Dordrecht: Reidel, 125–174.Google Scholar
- Judge, G. C., W. E. Griffiths, R. Carter-Hill, H. Lütkepohl, T.-C. Lee (1985). The Theory and Practice of Econometrics, New York: Wiley.Google Scholar
- Ley, E. (1992). “Applied Demand Analysis in Utility Space,” mimeo.Google Scholar
- Lindley, D. V. (1986). “Comment,” American Statistician, 40, 6–7.Google Scholar
- Osiewalski J. and M. F. J. Steel (1990). “A Bayesian Analysis of Exogeneity in Models Pooling Time-Series and Cross-Section Data,” mimeo.Google Scholar
- Raiffa, H. A. and R. S. Schlaifer (1961). Applied Statistical Decision Theory, Boston: Harvard University Press.Google Scholar
- Ripley, B. (1986). Stochastic Simulation, New York: Wiley.Google Scholar
- Smith, A. F. M. (1986). “Comment,” American Statistician, 40,10–11.Google Scholar
- van den Broeck, J., G. Koop, J. Osiewalski and M. F. J. Steel (1992). “Stochastic Frontier Models: A Bayesian Perspective,” Division de Economía WP 92–12, Universidad Carlos III de Madrid.Google Scholar