• Bernoulli
  • Volume 16, Number 3 (2010), 679-704.

Bayesian nonparametric estimation and consistency of mixed multinomial logit choice models

Pierpaolo De Blasi, Lancelot F. James, and John W. Lau

Full-text: Open access


This paper develops nonparametric estimation for discrete choice models based on the mixed multinomial logit (MMNL) model. It has been shown that MMNL models encompass all discrete choice models derived under the assumption of random utility maximization, subject to the identification of an unknown distribution G. Noting the mixture model description of the MMNL, we employ a Bayesian nonparametric approach, using nonparametric priors on the unknown mixing distribution G, to estimate choice probabilities. We provide an important theoretical support for the use of the proposed methodology by investigating consistency of the posterior distribution for a general nonparametric prior on the mixing distribution. Consistency is defined according to an L1-type distance on the space of choice probabilities and is achieved by extending to a regression model framework a recent approach to strong consistency based on the summability of square roots of prior probabilities. Moving to estimation, slightly different techniques for non-panel and panel data models are discussed. For practical implementation, we describe efficient and relatively easy-to-use blocked Gibbs sampling procedures. These procedures are based on approximations of the random probability measure by classes of finite stick-breaking processes. A simulation study is also performed to investigate the performance of the proposed methods.

Article information

Bernoulli, Volume 16, Number 3 (2010), 679-704.

First available in Project Euclid: 6 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian consistency blocked Gibbs sampler discrete choice models mixed multinomial logit random probability measures stick-breaking priors


De Blasi, Pierpaolo; James, Lancelot F.; Lau, John W. Bayesian nonparametric estimation and consistency of mixed multinomial logit choice models. Bernoulli 16 (2010), no. 3, 679--704. doi:10.3150/09-BEJ233.

Export citation


  • Antoniak, C.E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems., Ann. Statist. 2 1152–1174.
  • Barron, A., Schervish, M.J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems., Ann. Statist. 27 536–561.
  • Bhat, C. (1998). Accommodating variations in responsiveness to level-of-service variables in travel mode choice models., Transpn. Res. A 32 495–507.
  • Brownstone, D. and Train, K.E. (1999). Forecasting new product penetration with flexible substition patterns., J. Econometrics 89 109–129.
  • Cardell, N. and Dunbar, F. (1980). Measuring the societal impacts of automobile downsizing., Transpn. Res. A 14 423–434.
  • Choi, T. and Schervish, M.J. (2007). On posterior consistency in nonparametric regression problems., J. Multivariate Anal. 98 1969–1987.
  • Choudhuri, N., Ghosal, S. and Roy, A. (2005). Bayesian methods for function estimation. In, Handbook of Statistics (D. Dey, ed.) 25 377–418. Amsterdam: Elsevier.
  • Dubé, J.P., Chintagunta, P., Bronnenberg, B., Goettler, R., Petrin, A., Seetharaman, P.B., Sudhir, K., Thomadsen, R. and Zhao, Y. (2002). Structural applications of the discrete choice model., Marketing Letters 13 207–220.
  • Erdem, T. (1996). A dynamic analysis of market structure based on panel data., Marketing Science 15 359–378.
  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems., Ann. Statist. 1 209–230.
  • Ghosal, S., Ghosh, J.K. and Ramamoorthi, R.V. (1999). Posterior consistency of Dirichlet mixtures in density estimation., Ann. Statist. 27 143–158.
  • Ghosal, S. and Roy, A. (2006). Posterior consistency of Gaussian process prior for nonparametric binary regression., Ann. Statist. 34 2413–2429.
  • Ghosal, S. and Tang, Y. (2006). Bayesian consistency for Markov processes., Sankhyā 68 227–239.
  • Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors., J. Amer. Statist. Assoc. 96 161–173.
  • Ishwaran, H. and James, L.F. (2002). Approximate Dirichlet process computing in finite normal mixtures: Smoothing and prior information., J. Comp. Graph. Statist. 11 508–532.
  • Ishwaran, H. and Zarepour, M. (2000). Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchichal models., Biometrika 87 371–390.
  • James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measures with independent increments., Scand. J. Statist. 36 76–97.
  • Lijoi, A., Prünster, I. and Walker, S.G. (2005). On consistency of nonparametric normal mixtures for Bayesian density estimation., J. Amer. Statist. Assoc. 100 1292–1296.
  • Lo, A.Y. (1984). On a class of Bayesian nonparamertic estimates: I. Density estimates., Ann. Statist. 12 351–257.
  • MacEachern, S.N. and Muller, P. (1998). Estimating mixture of Dirichlet process models., J. Comput. Graph. Statist. 7 223–238.
  • McFadden, D. (1974). Conditional logit anaylsis of qualitative choice behavior. In, Frontiers of Econometrics (P. Zarembka, ed.) 105–142. New York: Academic Press.
  • McFadden, D. and Train, K.E. (2000). Mixed MNL models for discrete response., J. Appl. Econometrics 15 447–470.
  • Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator., Ann. Probab. 25 855–900.
  • Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of random measures with independent increments., Ann. Statist. 31 560–585.
  • Srinivasan, K. and Mahmassani, H. (2005). A dynamic kernel logit model for the analysis of longitude discrete choice data: Properties and computational assessment., Transportation Science 39 160–181.
  • Train, K.E. (2003)., Discrete Choice Methods with Simulation. Cambridge Univ. Press.
  • Train, K.E. (2008). EM algorithms for nonparametric estimation of mixing distributions., Journal of Choice Modelling 1 40–69.
  • Walker, J., Ben-Akiva, M. and Bolduc, D. (2007). Identification of parameters in normal error component logit-mixture (NECLM) models., J. Appl. Econometrics 22 1095–1125.
  • Walker, S.G. (2003a). On sufficient conditions for Bayesian consistency., Biometrika 90 482–488.
  • Walker, S.G. (2003b). Bayesian consistency for a class of regression problems., South African Statistist. J. 37 151–169.
  • Walker, S.G. (2004). New approaches to Bayesian consistency., Ann. Statist. 32 2028–2043.
  • Walker, S.G., Lijoi, A. and Prünster, I. (2005). Data tracking and the understanding of Bayesian consistency., Biometrika 92 765–778.
  • Wasserman, L. (1998). Asymptotic properties of nonparametric Bayesian procedures. In, Practical Nonparametric and Semiparametric Bayesian Statistics (D. Dey, P. Muller and D. Sinha, eds.) 293–304. New York: Springer-Verlag.