Abstract
Bayesian statistics is based up a philosophy different from that of other methods of statistical inference. In Bayesian statistics all unknowns, and in particular unknown parameters, are considered to be random variables and their probability distributions specify our beliefs about their likely values. Estimation, model selection, and uncertainty analysis are implemented by using Bayes’s theorem to update our beliefs as new data are observed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Edwards (1982) .
- 2.
OpenBUGS will run on a Mac under WINE.
- 3.
The effective sample size can be larger than the actual sample size if there is negative correlation, but negative correlation is unlikely. It is more likely that some of the effective sample sizes exceed the actual sample sizes due to random variation, i.e., estimation error.
- 4.
A residual is standardized by dividing it by its conditional standard deviation.
- 5.
JAGS had trouble when the full data set was used, probably because there are nearly 5000 latent variables. This problem is likely hardware dependent.
References
Albert, J. (2007) Bayesian Computation with R, Springer, New York.
Albert, J. H. and Chib, S. (1993) Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts, Journal of Business & Economic Statistics, 11, 1–15.
Berger, J. O. (1985) Statistical Decision Theory and Bayesian Analysis 2nd ed., Springer-Verlag, Berlin.
Bernardo, J. M., and Smith, A. F. M. (1994) Bayesian Theory, Wiley, Chichester.
Box, G. E. P., and Tiao, G. C. (1973) Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading, MA.
Brooks, S. P. and Gelman, A. (1998) General Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics, 7, 434–455.
Carlin, B. P., and Louis, T. A. (2000) Empirical Bayes: Past, present and future. Journal of the American Statistical Association, 95, 1286–1289.
Carlin, B., and Louis, T. A. (2008) Bayesian Methods for Data Analysis, 3rd ed., Chapman & Hall, New York.
Chib, S., and Ergashev, B. (2009) Analysis of multifactor affine yield curve models. Journal of the American Statistical Association, 104, 1324–1337.
Chib, S., and Greenberg, E. (1994) Bayes inference in regression models with ARMA(p, q) errors. Journal of Econometrics, 64, 183–206.
Chib, S., and Greenberg, E. (1995) Understanding the Metropolis–Hastings algorithm. American Statistician, 49, 327–335.
Congdon, P. (2001) Bayesian Statistical Modelling, Wiley, Chichester.
Congdon, P. (2003) Applied Bayesian Modelling, Wiley, Chichester.
Daniels, M. J., and Kass, R. E. (1999) Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. Journal of the American Statistical Association, 94, 1254–1263.
Edwards, W. (1982) Conservatism in human information processing. In Judgement Under Uncertainty: Heuristics and Biases, D. Kahneman, P. Slovic, and A. Tversky, ed., Cambridge University Press, New York.
Gelman, A., and Rubin, D. B. (1992) Inference from iterative simulation using multiple sequence (with discussion). Statistical Science, 7, 457–511.
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013) Bayesian Data Analysis, 3rd ed., Chapman & Hall, London.
Greyserman, A., Jones, D. H., and Strawderman, W. E. (2006) Portfolio selection using hierarchical Bayesian analysis and MCMC methods, Journal of Banking and Finance, 30, 669–678.
Kass, R. E., Carlin, B. P., Gelman, A., and Neal, R. (1998) Markov chain Monte Carlo in practice: A roundtable discussion. American Statistician, 52, 93–100.
Kim, S., Shephard, N., and Chib, S. (1998) Stochastic volatility: likelihood inference and comparison with ARCH models.Review of Economic Studies, 65, 361–393.
Ledoit, O., and Wolf, M. (2003) Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10, 603–621.
Lehmann, E. L. (1983) Theory of Point Estimation, Wiley, New York.
Lunn, D., Jackson, C., Best, N., Thomas, A., and Spiegelhalter, D. (2013) The BUGS Book, Chapman & Hall.
Lunn, D. J., Thomas, A., Best, N., and Spiegelhalter, D. (2000) OpenBUGS—A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.
Rachev, S. T., Hsu, J. S. J., Bagasheva, B. S., and Fabozzi, F. J. (2008) Bayesian Methods in Finance, Wiley, Hoboken, NJ.
Robert, C. P. (2007) The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, 2nd ed., Springer, New York.
Robert, C. P., and Casella, G. (2005) Monte Carlo Statistical Methods, 2nd ed., Springer, New York.
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002) Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, Methodological, 64, 583–616.
Stein, C. (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical and Statistical Probability, J. Neyman, ed., University of California, Berkeley, pp. 197–206, Volume 1.
van der Vaart, A. W. (1998) Asymptotic Statistics, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ruppert, D., Matteson, D.S. (2015). Bayesian Data Analysis and MCMC. In: Statistics and Data Analysis for Financial Engineering. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2614-5_20
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2614-5_20
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2613-8
Online ISBN: 978-1-4939-2614-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)