Models of consumer heterogeneity play a pivotal role in marketing and economics, specifically in random coefficient or mixed logit models for aggregate or individual data and in hierarchical Bayesian models of heterogeneity. In applications, the inferential target often pertains to a population beyond the sample of consumers providing the data. For example, optimal prices inferred from the model are expected to be optimal in the population and not just optimal in the observed, finite sample. The population model, random coefficients distribution, or heterogeneity distribution is the natural and correct basis for generalizations from the observed sample to the market. However, in many if not most applications standard heterogeneity models such as the multivariate normal, or its finite mixture generalization lack economic rationality because they support regions of the parameter space that contradict basic economic arguments. For example, such population distributions support positive price coefficients or preferences against fuel-efficiency in cars. Likely as a consequence, it is common practice in applied research to rely on the collection of individual level mean estimates of consumers as a representation of population preferences that often substantially reduce the support for parameters in violation of economic expectations. To overcome the choice between relying on a mis-specified heterogeneity distribution and the collection of individual level means that fail to measure heterogeneity consistently, we develop an approach that facilitates the formulation of more economically faithful heterogeneity distributions based on prior constraints. In the common situation where the heterogeneity distribution comprises both constrained and unconstrained coefficients (e.g., brand and price coefficients), the choice of subjective prior parameters is an unresolved challenge. As a solution to this problem, we propose a marginal-conditional decomposition that avoids the conflict between wanting to be more informative about constrained parameters and only weakly informative about unconstrained parameters. We show how to efficiently sample from the implied posterior and illustrate the merits of our prior as well as the drawbacks of relying on means of individual level preferences for decision-making in two illustrative case studies.

消费者异质性模型在营销和经济学中发挥着关键作用，特别是在总体或个体数据的随机系数或混合逻辑模型以及异质性的分层贝叶斯模型中。在应用中，推理目标通常涉及超出提供数据的消费者样本的人群。例如，从模型推断出的最优价格预计在总体中是最优的，而不仅仅是在观察到的有限样本中最优。总体模型、随机系数分布或异质性分布是从观察样本到市场的概括的自然且正确的基础。然而，在许多（如果不是大多数）应用中，标准异质性模型（例如多元正态模型或其有限混合泛化）缺乏经济合理性，因为它们支持与基本经济论据相矛盾的参数空间区域。例如，这种人口分布支持正价格系数或对汽车燃油效率的偏好。因此，应用研究中的常见做法是依靠收集消费者的个人水平平均估计来代表人口偏好，这通常会大大减少对参数的支持，从而违反经济预期。为了克服依赖错误指定的异质性分布和收集个体水平意味着无法一致地测量异质性之间的选择，我们开发了一种方法，有助于根据先验约束制定更经济可靠的异质性分布。在异质性分布同时包含受约束和无约束系数（例如，品牌和价格系数）的常见情况下，主观先验参数的选择是一个未解决的挑战。作为这个问题的解决方案，我们提出了一种边际条件分解，避免了想要获得更多有关受约束参数的信息与仅获得有关不受约束参数的弱信息之间的冲突。我们展示了如何有效地从隐含后验中进行采样，并在两个说明性案例研究中说明了先验的优点以及依赖个人水平偏好手段进行决策的缺点。