Bayesian model criticism

Bayesian Model criticism

Prior and posterior model probabilities, marginal likelihood and Bayes factor.  Computing the normalizing constant p(y) – Laplace-Metropolis estimator, simple Monte Carlo estimator, Monte Carlo estimator via importance sampling, annealed importance sampling estimator, brigde sampling estimator, path sampling estimator, Chib’s estimator and Chib, Jeliazkov’s estimator, and Savage-Dickey density ratio.  Trans-dimensional MCMC algorithms – Green’s (1995) RJMCMC, Carlin and Chib’s (1995) pseudo-priors, Godsill’s (2001) partial analytic RJMCMC and Dellaportas et al’s (2002) Metropolized Carlin-Chib.  Bayesian model averaging (BMA). Deviance information criterion (DIC)

Additional material

• The binomial regression revisited (R code)
• A probit regression (R code)
• Bayesian multiple linear regression (R code) + (college.txt)
• A note on the Reversible jump MCMC algorithm (by Hedibert Lopes)
• Bayesian Model Averaging website (by Chris Volinsky)

A few additional references

• Newton, M. A. and Raftery, A. E. (1994) Approximate Bayesian inference by the weighted likelihood bootstrap (with discussion). JRSS-B 56, 3-48.
• Chib, S. (1995) Marginal likelihood from the Gibbs output. JASA, 90, 773-95.
• Kass, R. E. and Raftery, A. E.(1995) Bayes factors. JASA, 90, 773-95.
• Meng, X. L. and Wong, W. H.(1996) Simulating ratios of normalizing constants via a simple identity: a theoretical exploration. Statistica Sinica, 6, 831-60.
• DiCiccio, T. J., Kass, R. E., Raftery, A. E. and Wasserman, L. (1997) Computing Bayes factors by combining simulation and asymptotic approximations. JASA, 92, 903-15.
• Gelman, A. and Meng, X. L.(1998) Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statistical Science, 13, 163-85.
• Gelfand, A. E. and Dey, D. K. (1994) Bayesian model choice: asymptotics and exact calculations. JRSS-B, 56, 501-14.
• Chib, S. and Jeliazkov, I. (2001) Marginal likelihood from the Metropolis-Hastings output. JASA, 96, 270-81.
•  Neal, R. M. (2001) Annealed importance sampling. Statistics and Computing, 11, 125-39.
• Carlin, B. P. and Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo methods. JRSS-B, 57, 473-84.
• Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711-32.
• Verdinelli, I. and Wasserman, L. (1995) Computing Bayes factor using a generalization of the Savage-Dickey density ratio, JASA, 90, 614-8.
• Raftery, Madigan and Hoeting (1997) Bayesian Model Averaging for Linear Regression Models, JASA, 92, 179-191.
• Hoeting, Madigan, Raftery and Volinsky (1999) Bayesian Model Averaging, Statistical Science, 14, 382-401.
• Godsill, S. J. (2001) On the relationship between Markov chain Monte Carlo methods for model uncertainty. JCGS, 10, 1-19.
• Dellaportas, P., Forster, J. and Ntzoufras, I. (2002) On Bayesian model and variable selection using MCMC. Statistics and Computing, 12, 27-36.
• Spiegelhalter, D. J., Best, N.G., Carlin, B.P., and van der Linde, A. (2002) Bayesian measures of model complexity and fit (with discussion and rejoinder), JRSS-B, 64, 583-639.
• Spiegelhalter, D. J., Best, N.G., Carlin, B.P., and van der Linde, A. (2014) The deviance information criterion: 12 years on, JRSS-B, 73(3), 485,493.
• Koop (2003) Bayesian Econometrics, Wiley, Chapter 13: “Bayesian Model Averaging” gives a detailed introduction of BMA in linear regression models.
• Lopes, H. F. and West, M. (2004) Bayesian Model Assessment in Factor Analysis. Statistica Sinica, 14, 41-67.
• van der Linde, A. (2005) DIC in variable selection. Statistica Neerlandica, 59, 45-56.