## 1 Introduction 

In 1970, John D. C. Little famously wrote: “The big problem with management science models is that managers practically never use them. There have been a few applications, of course, but the practice is a pallid picture of the promise” (Little 1970). The same may be true today about Bayesian estimation of hierarchical probability models. The impact Bayesian methods have had on academic research across multiple disciplines in the managerial, social and natural sciences is undeniable. Marketing, in particular, has benefited from Bayesian methods because of their natu- ral suitability for capturing heterogeneity in customer types and tastes (Rossi and Allenby 2003). But further diffusion of Bayesian methods is constrained by a scalability problem. As the size and complexity of data sources for both research and commercial purposes grows, the impracticality of simulation-based Bayesian methods for estimating parameters of a general class of hierarchical models becomes increasingly salient (Allenby et al. 2014). 

The problem is not with the Bayesian approach itself, but with the most familiar methods of simulating from the posterior distributions of the parameters. Without question, the resurgence of Bayesian ideas is due to the popularity of Markov chain Monte Carlo (MCMC), which was introduced to statistical researchers by Gelfand and Smith (1990) via the Gibbs sampler. MCMC estimation involves iteratively sampling from the marginal posterior distributions of blocks of parameters. Only after some unknown (and theoretically infinite) number of iterations will the algorithm generate samples from the correct distributions; earlier samples are discarded. The Bayesian computational literature has exploded with numerous methods for generating valid and efficient MCMC algorithms. It would be difficult to list all of them here, so we refer the reader to Gelman et al. (2003); Chen et al. (2000); Rossi et al. (2005) and Brooks et al. (2010), and the hundreds of references therein. 

Despite the justifiable success MCMC has enjoyed, there remains the question of whether a particular chain has run long enough that we can start collecting samples for estimation (or, collo- quially, whether the chain has “converged” to the target distribution). This is a particular problem for hierarchical models in which each heterogeneous unit is characterized by its own set of pa- rameters. For example, each household in a customer dataset might have its own preferences for product attributes. Both the number of parameters and the cycle time for each MCMC iteration 

