Heterogeneity in HMMs: Allowing for Heterogeneity in the Number of States

Netzer, O, Padilla, N and Montoya, R (2019) Heterogeneity in HMMs: Allowing for Heterogeneity in the Number of States. Working Paper. Columbia Business School.

Abstract

Hidden Markov Models (HMMs) have been widely used in marketing to study dynamics in customer behavior. HMMs have been successfully applied to model how customers transition among a set latent states such as attention levels, web search behavior, customer's relationships, and purchase intent. While most HMMs in marketing allow for heterogeneity in the model's parameters, these models assume that the number of latent states is common across customers. In this work, we analyze the potential bias of making such an assumption, assess to what extent heterogeneity in the model's parameters can mitigate the impact of such bias, and provide a mixture of HMMs model that relaxes this assumption. Using a comprehensive Monte Carlo simulation exercise and secondary data from an online role playing game, we demonstrate that ignoring heterogeneity in the number of states could lead to model identification problems and to erroneous interpretations of customer dynamics. In particular, we show that: (1) even when only a small proportion of customers have a larger number of latent states (and most customers transition among fewer states), the best fitting model would be an expensive HMM in terms of number of states; (2) even when heterogeneity is accounted for in the HMM parameters, the inference from analyzing the population estimates, a common practice in the literature, can be biased; (3) even the individual-level estimates of customers with the correct number of states can be biased. We propose a mixture of HMMs with different number of states to account for heterogeneity in the number of states which captures well the behavior at both individual and population level.