Bayesian Inference in the Multinomial Logit Model


  • Sylvia Frühwirth-Schnatter University of Economics and Business, Vienna
  • Rudolf Frühwirth Austrian Academy of Sciences, Vienna



The multinomial logit model (MNL) possesses a latent variable representation in terms of random variables following a multivariate logistic distribution. Based on multivariate finite mixture approximations of the multivariate logistic distribution, various data-augmented Metropolis-Hastings algorithms are developed for a Bayesian inference of the MNL model.


Albert, J. H., and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.

Alspach, D. L., and Sorenson, H. W. (1972). Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Transactions on Automatic Control, 17, 439–448.

Balakrishnan, N. (1992). Handbook of the Logistic Distribution. New York: Marcel Dekker.

Chib, S., Nardari, F., and Shephard, N. (2002). Markov chain Monte Carlo methods for stochastic volatility models. Journal of Econometrics, 108, 281–316.

Fahrmeir, L., and Tutz, G. (2001). Multivariate Statistical Modelling based on Generalized Linear Models (2nd ed.). New York/Berlin/Heidelberg: Springer.

Fox, J. (2010). Bayesian Item Response Modeling. New York: Springer.

Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. New York: Springer.

Frühwirth-Schnatter, S., and Frühwirth, R. (2007). Auxiliary mixture sampling with applications to logistic models. Computational Statistics and Data Analysis, 51, 3509–3528.

Frühwirth-Schnatter, S., and Frühwirth, R. (2010). Data augmentation and MCMC for binary and multinomial logit models. In T. Kneib and G. Tutz (Eds.), Statistical Modelling and Regression Structures – Festschrift in Honour of Ludwig Fahrmeir (pp. 111–132). Heidelberg: Physica-Verlag. (Also available at, IFAS Research Paper Series 2010-48)

Frühwirth-Schnatter, S., Frühwirth, R., Held, L., and Rue, H. (2009). Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. Statistics and Computing, 19, 479-492.

Frühwirth-Schnatter, S., and Wagner, H. (2006). Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling. Biometrika, 93, 827–841.

Gamerman, D., and Lopes, H. F. (2006). Markov Chain Monte Carlo. Stochastic Simulation for Bayesian Inference (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.

Geyer, C. (1992). Practical Markov chain Monte Carlo. Statistical Science, 7, 473–511.

Gumbel, E. J. (1961). Bivariate logistic distributions. Journal of the American Statistical Association, 56, 335–349.

Guttman, I., Dutter, R., and Freeman, P. R. (1978). Care and handling of univariate outliers in the general linear model to detect spuriosity — A Bayesian approach. Technometrics, 20, 187–193.

Holmes, C. C., and Held, L. (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis, 1, 145-168.

Imai, K., and van Dyk, D. A. (2005). A Bayesian analysis of the multinomial probit model using marginal data augmentation. Journal of Econometrics, 124, 311–334.

Kass, R. E., Carlin, B., Gelman, A., and Neal, R. (1998). Markov chain Monte Carlo in practice: A roundtable discussion. The American Statistician, 52, 93–100.

Kotz, S., Johnson, N. L., and Balakrishnan, N. (2000). Continous Multivariate Distributions: Models and Applications. Wiley.

Liu, C. (2004). Robit regression: a simple robust alternative to logistic and probit regression. In A. Gelman and X.-L. Meng (Eds.), Applied Bayesian Modeling and Casual Inference from Incomplete-Data Perspectives (pp. 227–238). Chichester: Wiley.

Malik, H. J., and Abraham, B. (1973). Multivariate logistic distributions. The Annals of Statistics, 1, 588–590.

McCulloch, R. E., Polson, N. G., and Rossi, P. E. (2000). A Bayesian analysis of the multinomial probit model with fully identified parameters. Journal of Econometrics, 99, 173–193.

McFadden, D. (1974). Conditional logit analysis of qualitative choice behaviour. In P. Zarembka (Ed.), Frontiers of Econometrics (pp. 105–142). New York: Academic.

Nelder, J. A., and Mead, R. (1965). A Simplex Method for Function Minimization. Computer Journal, 7, 308–313.

Newcomb, S. (1886). A generalized theory of the combination of observations so as to obtain the best result. American Journal of Mathematics, 8, 343–366.

Omori, Y., Chib, S., Shephard, N., and Nakajima, J. (2007). Stochastic volatility with leverage: Fast and efficient likelihood inference. Journal of Econometrics, 140, 425–449.

Rossi, P. E., Allenby, G. M., and McCulloch, R. (2005). Bayesian Statistics and Marketing. Chichester: Wiley.

Scott, S. L. (2011). Data augmentation, frequentist estimation, and the Bayesian analysis of multinomial logit models. Statistical Papers, 52, 87–109.

Shephard, N. (1994). Partial non-Gaussian state space. Biometrika, 81, 115–131.

Sorenson, H. W., and Alspach, D. L. (1971). Recursive Bayesian estimation using Gaussian sums. Automatica, 6, 465–479.

Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions. New York: Wiley.

van Dyk, D. A., and Park, T. (2008). Partially collapsed Gibbs samplers: Theory and methods. Journal of the American Statistical Association, 103, 790–796.




How to Cite

Frühwirth-Schnatter, S., & Frühwirth, R. (2016). Bayesian Inference in the Multinomial Logit Model. Austrian Journal of Statistics, 41(1), 27–43.