Modelling and Prediction Honoring Seymour Geisser pp 206-227 | Cite as

# Orthogonalizations and Prior Distributions for Orthogonalized Model Mixing

## Abstract

Prediction methods based on mixing over a set of plausible models can help alleviate the sensitivity of inference and decisions to modeling assumptions. One important application area is prediction in linear models. Computing techniques for model mixing in linear models include Markov chain Monte Carlo methods as well as importance sampling. Clyde, DeSimone and Parmigiani (1996) developed an importance sampling strategy based on expressing the space of predictors in terms of an orthogonal basis. This leads both to a better identified problem and to simple approximations to the posterior model probabilities. Such approximations can be used to construct efficient importance samplers. For brevity, we call this strategy orthogonalized model mixing.

Two key elements of orthogonalized model mixing are: a) the orthogonalization method and b) the prior probability distributions assigned to the models and the coefficients. In this paper we consider in further detail the specification of these two elements. In particular, after identifying the aspects of these specifications that are essential to the success of the importance sampler, we list and briefly discuss a number of different alternatives for both a) and b). We highlight the features that may make each one of the options attractive in specific situations and we illustrate some important points via a simulated data set.

### Keywords

Covariance Shrinkage Stein Oman Parmi## Preview

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### References

- Bernardo, J.M. and Smith A.F.M. (1994)
*Bayesian Theory*. Wiley, N.YCrossRefMATHGoogle Scholar - Chipman, H (1996). Bayesian Variable Selection with Related Predictors.
*Canadian Journal of Statistics*to appearGoogle Scholar - Clyde, MA, DeSimone, H, and Parmigiani, G (1996). Prediction via Orthogonalized Model Mixing.
*Journal of the American Statistical Association*, forthcomingGoogle Scholar - Draper, D (1995). Assessment and propagation of model uncertainty (with Discussion) .
*Journal of the Royal Statistical Society*57, pp. 45–98MATHMathSciNetGoogle Scholar - Foster, D, George, E and McCulloch, R (1995) Calibrating Bayesian variable selection proceduresGoogle Scholar
- Garthwaite, PH and Dickey, JM (1992). Elicitation of prior distributions for variable selection problems in regression.
*Annals of Statistics*20, pp. 1697–1719Google Scholar - Geisser, S, Predictive inference. An introduction, Chapman & Hall, New York, 1993Google Scholar
- George, E.I. (1986). Minimax multiple shrinkage estimation.
*Annals of Statistics*14, pp. 188–205CrossRefMATHMathSciNetGoogle Scholar - George, EI (1986). Combining minimax shrinkage estimators.
*Journal of the American Statistical Association*81, pp. 437–445CrossRefMATHMathSciNetGoogle Scholar - George, EI and McCulloch, R (1993). Variable Selection via Gibbs Sampling.
*Journal of the American Statistical Association*88, pp. 881–889CrossRefGoogle Scholar - George, EI and McCulloch, R (1994). Fast Bayes Variable Selection. TR, Graduate School of Business, University of ChicagoGoogle Scholar
- George, E.I. and Oman, S.D. (1993). Multiple shrinkage principal component regression. TR, University of Texas at AustinGoogle Scholar
- Geweke, JF (1994). Bayesian comparison of econometric models. Working Paper 532, Federal Reserve Bank of MinneapolisGoogle Scholar
- Green, PJ (1995). Reversible Jump MCMC Computation and Bayesian Model determination. TR-94-19, University of Bristol.Google Scholar
- Hoeting, J, Raftery, AE, Madigan, DM (1995), A Method for Simultaneous Variable Selection and Outlier Identification in Linear Regression, Technical Report 95–02, Colorado State UniversityGoogle Scholar
- Jolliffe, I.T. (1986) Principal Component Analysis, Springer, New YorkCrossRefGoogle Scholar
- Jolliffe, I.T. (1982). A note on the use of principal components in regression.
*Applied Statistics*31, pp. 300–303CrossRefGoogle Scholar - Kadane, J.B., Dickey, J.M., Winkler, R.L., Smith, W.S., and Peters, S.C. (1980). Interactive elicitation of opinion for a normal linear model.
*Journal of the American Statistical Association*75, pp. 845–85CrossRefMathSciNetGoogle Scholar - Laud, P and Ibrahim, JG (1994). Predictive Specification of Prior Model Probability in Variable Selection. TR, Division of Statistics, University of Northern IllinoisGoogle Scholar
- Li, K-C. (1991). Sliced inverse regression for dimension reduction.
*Journal of the American Statistical Association*86, pp. 316–327CrossRefMATHMathSciNetGoogle Scholar - Mitchell, T.J. and Beauchamp, J.J. (1988). Bayesian Variable Selection in Linear Regression.
*Journal of the American Statistical Association*83, pp. 1023–1036CrossRefMATHMathSciNetGoogle Scholar - Palm F.C. and Zellner A. (1992). “To Combine or not to Combine? Issues of Combining Forecasts,”
*Journal of Forecasting*, 11, 687–701CrossRefGoogle Scholar - Phillips, D.B. and Smith, A.F.M. (1994). Bayesian Model Comparison via Jump Diffusions. TR-94-20, Department of Mathematics, Imperial College, UKGoogle Scholar
- Raftery, AE, Madigan, DM, and Hoeting, J (1993). Model selection and accounting for model uncertainty in linear regression models. TR 262, Department of Statistics, University of WashingtonGoogle Scholar
- Raftery, A.E., Madigan, D.M., and Volinski C.T. (1995). Accounting for Model Uncertainty in Survival Analysis Improves Predictive Performance (with discussion). In
*Bayesian Statistics*5, ed. J.M. Bernardo, J.O. Berger, A.P. Dawid and Smith, A.F.MGoogle Scholar - Rao, C.R. (1964). The Use and Interpretation of Principal Components in Applied Research.
*Sankhya A*26, pp. 329–358MATHGoogle Scholar - Stone, M. and Brooks, R. J. (1990). Continuum regression: Cross-validated sequentially constructed prediction embracing ordinary least squares, partial least squares and principal components regression.
*Journal of the Royal Statistical Society, Series B*52, pp. 237–269MATHMathSciNetGoogle Scholar - Weisberg, S (1985).
*Applied Linear Regression. 2nd Edition*. Wiley, New York.Google Scholar - Wold, S., Ruhe, A., Wold, H., and Dunn, W.J.III (1984). The collinearity problem in linear regression. The partial least squares (PLS) approach to generalized inverses.
*SI AM Journal on Scientific and Statistical Computing*5, pp. 735–743CrossRefMATHGoogle Scholar - Zellner, A., (1986). On assessing prior distribution in Bayesian regression analysis with
*g-*prior distributions. In “Bayesian Inference and decision Techniques: Essays in Honor of Bruno de Finetti”, 233–243, ( P.K. Goel and A. Zellner eds.). North Holland, AmsterdamGoogle Scholar