The finite normal mixture model has emerged as a dominant methodology for assessing heterogeneity in choice models. Although it extends the classic mixture models by allowing within component variablility, it requires that a relatively large number of models be separately estimated and fairly difficult test procedures to determine the "correct" number of mixing components. We present a very general formulation, based on Dirichlet Process Piror, which yields the number and composition of mixing components a posteriori, obviating the need for post hoc test procedures and is capable of approximating any target heterogeneity distribution. Adapting Stephens' (2000) algorithm allows the determination of 'substantively' different clusters, as well as a way to sidestep problems arising from label-switching and overlapping mixtures. These methods are illustrated both on simulated data and A.C. Nielsen scanner panel data for liquid detergents. We find that the large number of mixing components required to adequately represent the heterogeneity distribution can be reduced in practice to a far smaller number of segments of managerial relevance.

Empirical evidence suggests that decision makers often weight successive additional units of a valued attribute or monetary endowment unequally, so that their utility functions are intrinsically nonlinear or irregularly shaped. Although the analyst may impose various functional specifications exogenously, this approach is ad hoc, tedious, and reliant on various metrics to decide which specification is "best." In this paper, we develop a method that yields individual-level, flexibly shaped utility functions for use in choice models. This flexibility at the individual level is accomplished through splines of the truncated power basis type in a general additive regression framework for latent utility. Because the number and location of spline knots are unknown, we use the birth-death process of Denison et al. (1998) and Green's (1995) reversible jump method. We further show how exogenous constraints suggested by theory, such as monotonicity of price response, can be accommodated. Our formulation is particularly suited to estimating reaction to pricing, where individual-level monotonicity is justified theoretically and empirically, but linearity is typically not. The method is illustrated in a conjoint application in which all covariates are splined simultaneously and in three panel data sets, each of which has a single price spline. Empirical results indicate that piecewise linear splines with a modest number of knots fit these data well, substantially better than heterogeneous linear and log-linear a priori specifications. In terms of price response specifically, we find that although aggregate market-level curves can be nearly linear or log-linear, individuals often deviate widely from either. Using splines, hold-out prediction improvement over the standard heterogeneous probit model ranges from 6% to 14% in the scanner applications and exceeds 20% in the conjoint study. Moreover, "optimal" profiles in conjoint and aggregate price response curves in the scanner applications can differ markedly under the standard and the spline-based models.choice models, utility theory, heterogeneity, splines, Bayesian methods, Markov chain Monte Carlo

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