[01] Azad Abdulhafedh, Department of Civil Engineering, University of Missouri, Columbia, Missouri, USA.

Bayesian and frequentist analyses are two fundamental approaches to statistical modeling. Along with the rapid development of the frequentist analysis, Bayesian methodology was also developing, although with less attention and at a slower pace. One obstacle for the progress of Bayesian analysis has been the lack of adequate computational resources. With the emerging advances in statistical and computational software’s, the Bayesian analysis is currently widely accepted by researchers and practitioners as a feasible alternative. Bayesian statistics focuses on the probability of the hypothesis, given the data. Frequentist statistics focuses on the probability of the data, given the hypothesis. In addition, Bayesian and frequentist approaches have different views about what is considered fixed and random, and therefore, have different interpretations of the outcomes. The Bayesian analysis assumes that the observed data sample is fixed, model parameters are random, and the posterior distribution of parameters is estimated based on the observed data and the prior distribution of parameters. The frequentist analysis assumes that the observed data is a repeatable random sample and that parameters are fixed across the repeated samples. This paper provides an overview to better understand the Bayesian analysis compared to the frequentist analysis.

Bayesian Analysis, Frequentist Analysis, Posterior Distribution, Markov Chain Monte Carlo

[01] Brooks, S. A., Gelman, G. L., Jones, G. L., & Meng, X. H. (2011). Handbook of Markov Chain Monte Carlo. FL, USA: Chapman & Hall-CRC. [02] Xu, C., Wang, W., Liu, P., & Li, Z. (2014). Using the Bayesian updating approach to improve the spatial and temporal transferability of real-time crash risk prediction models. Accident Analysis and Prevention, 38, 167–176. [03] Lord, D., & Miranda-Moreno, F. (2008). Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poisson-gamma models for modeling motor vehicle crashes: A Bayesian perspective. Accident Analysis and Prevention, 46 (5), 751–770. [04] Miaou, P., & Song, J. (2005). Bayesian ranking of sites for engineering safety improvements: decision parameter, treatability concept, statistical criterion and spatial dependence. Accident Analysis and Prevention, 37 (4), 699–720. [05] MacNab, Y. C. (2004). Bayesian spatial and ecological models for small-area accident and injury analysis. Accident Analysis and Prevention, 36 (6), 1028–1091. [06] Aguero-Valverde, J., & Jovanis, P. (2006). Spatial analysis of fatal and injury crashes in Pennsylvania. Accident Analysis and Prevention, 38 (3), 618–625. [07] Huang H., Darwiche, A. L., & Abdel-Aty, M. A. (2010). County-Level Crash Risk Analysis in Florida: Bayesian Spatial Modeling. Transportation Research Record, 2148, 27–37. [08] Yu, R., Abdel-Aty, M., & Ahmed, M. (2013). Bayesian random effect models incorporating real-time weather and traffic data to investigate mountainous freeway hazardous factors. Accident Analysis and Prevention, 50, 371–376. [09] Metropolis, N., Rosenbluth, A. W., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092. [10] Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741. [11] Geweke, J. (1999). Using simulation methods for Bayesian econometric models: Inference, development, and communication. Econometric Reviews, 18, 1–73. [12] Chernozhukov, V., & Hong, H. (2003). An MCMC approach to classical estimation. Journal of Econometrics, 115, 293–346. [13] Giordani, P., & Kohn, J. (2010). Adaptive independent Metropolis–Hastings by fast estimation of mixtures of normals. Journal of Computational and Graphical Statistics, 19, 243–259. [14] Gelman, Andrew, Jessica Hwang, & Aki Vehtari. (2014). “Understanding Predictive Information Criteria for Bayesian Models.” Statistics and Computing 24 (6). Springer: 997–1016. [15] Elster, C. & G. Wübbeler. (2015). “Bayesian regression versus application of least squares-An example,” Metrologia, vol. 53, no. 1, pp. S10–S16. [16] Ben Said, A., M. K. Shahzad, E. Zamai, S. Hubac, & M. Tollenaere. (2016). “Experts’ knowledge renewal and maintenance actions effectiveness in high-mix low-volume industries, using Bayesian approach,” Cognition, Technology & Work, vol. 18, no. 1, pp. 193–213. [17] Bernardo, J. M., & Smith, M. (2000). Bayesian Theory. Chichester, UK: Wiley. [18] Eberly, L. E., & Casella, G. (2003). Estimating Bayesian credible intervals. Journal of Statistical Planning and Inference, 112, 115–132. [19] Carlin, B. P., & Louis, A. (2000). Bayes and Empirical Bayes Methods for Data Analysis (2nd ed). Boca Raton, FL: Chapman & Hall-CRC. [20] Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2003). Bayesian Data Analysis (2nd ed). London, UK: Chapman and Hall, CRC. [21] Tanner, M. A. (1996). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions (3rd ed.). New York: Springer. [22] Tanner, M. A., & Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). Journal of the American Statistical Association, 82, 528–550. [23] Bolstad, W. M., (2007). Introduction to Bayesian Statistics: Second Edition, John Wiley. [24] Carlin, B. P., & Louis, A. (2000). Bayes and Empirical Bayes Methods for Data Analysis (2nd ed.). Boca Raton, FL: Chapman & Hall-CRC. [25] Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795. [26] Pollard, W. E. (1986). Bayesian Statistics for Evaluation Research: An Introduction. Newbury Park, CA: Sage Publications. [27] Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, 22, 1701–1728. [28] Gelman, A., & Rubin, D. B. (1992). Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7, 457–472. [29] Geweke, J. (1992). Evaluating the Accuracy of Sampling-Based Approaches to Calculating Posterior Moments. Oxford, UK: Clarendon Press. [30] Heidelberger, P., & Welch, D. (1983). Simulation Run Length Control in the Presence of an Initial Transient. Operations Research, 31, 1109–1144. [31] Raftery, A. E., & Lewis, S. M. (1992). One Long Run with Diagnostics: Implementation Strategies for Markov Chain Monte Carlo. Statistical Science, 7, 493–497.