UCL School of Management is delighted to welcome Thomas Otter, Goethe, to host a research seminar discussing “How to Generalize from a Hierarchical Model?”
In many marketing applications of hierarchical models the goal is to inform actions that apply to the population of consumers beyond the sample available for calibration; the goal is to generalize to the population, an exercise often referred to as market simulation. Examples are price and product optimization based on household scanner panel data, or data from discrete choice experiments. In a hierarchical model, the posterior of the hierarchical prior is the natural basis for such generalizations.
However, in many if not most marketing applications standard hierarchical prior distributions such as the multivariate normal, or its finite mixture generalization should be considered mis-specified because they support regions of the parameter space that contradict basic economic arguments.
For example, such priors support positive price coefficients or preferences against fuel-efficiency in cars. Likely as a consequence, it is common practice to rely on the collection of individual level posterior mean preferences of in-sample respondents, or consumers, as a representation of population preferences. Shrinkage of individual level posterior means towards the population mean often substantially reduces the support for parameters in violation of economic expectations in this collection compared to the posterior of the hierarchical prior, at the cost of inconsistent inference for preference heterogeneity. To overcome the choice between relying on the posterior of a mis-specified hierarchical prior and the collection of individual level posterior means that fail to measure heterogeneity, we show how to specify more economically faithful hierarchical prior distributions based on prior constraints. We develop our prior decomposing the hierarchical prior into a more informative marginal prior for parameters relating to constraints and a weakly informative, “barely proper” conditional prior for the remaining parameters. We show how to efficiently sample from the implied posterior and demonstrate practical relevance in two empirical case studies.