Hamiltonian sequential Monte Carlo with application to consumer choice behavior

AbstractThe practical use of nonparametric Bayesian methods requires the availability of efficient algorithms for posterior inference. The inherently serial nature of traditional Markov chain Monte Carlo (MCMC) methods imposes limitations on their efficiency and scalability. In recent years, there has been a surge of research activity devoted to developing alternative implementation methods that target parallel computing environments. Sequential Monte Carlo (SMC), also known as a particle filter, has been gaining popularity due to its desirable properties. SMC uses a genetic mutation-selection sampling approach with a set of particles representing the posterior distribution of a stochastic process. We propose to enhance the performance of SMC by utilizing Hamiltonian transition dynamics in the particle transition phase, in place of random walk used in the previous literature. We call the resulting procedure Hamiltonian Sequential Monte Carlo (HSMC). Hamiltonian transition dynamics have been shown to yield superior mixing and convergence properties relative to random walk transition dynamics in the context of MCMC procedures. The rationale behind HSMC is to translate such gains to the SMC environment. HSMC will facilitate practical estimation of models with complicated latent structures, such as nonparametric individual unobserved heterogeneity, that are otherwise difficult to implement. We demonstrate the behavior of HSMC in a challenging simulation study and contrast its favorable performance with SMC and other alternative approaches. We then apply HSMC to a panel discrete choice model with nonparametric consumer heterogeneity, allowing for multiple modes, asymmetries, and data-driven clustering, providing insights for consumer segmentation, individual level marketing, and price micromanagement.Keywords: Bayesian nonparametricsdiscrete choicemixed panel logitparticle filteringJEL Classification: C11C14C15C23C25 AcknowledgmentsWe would like to thank the Editor, two anonymous referees, and the participants of the 11th International Conference on Computational and Financial Econometrics (CFE), University of London, UK, 2017, and the European Seminar on Bayesian Econometrics (ESOBE), New Orleans, USA, 2018, for insightful comments and suggestions. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium.Notes1 Recent trends include shared memory multicore CPUs, GPUs, and distributed memory high-performance clusters.2 For the sake of simplicity in the notation of this section, we suppress the dependence of the posterior on data and other parameters not directly sampled.3 In the physics literature, θ denotes the position (or state) variable and − ln π(θ) describes its potential energy, while γ is the momentum variable with kinetic energy γ′M−1γ/2, yielding the total energy H(θ,γ) of the system, up to a constant of proportionality. M is a constant, symmetric, positive-definite “mass” matrix which is often set as a scalar multiple of the identity matrix.4 In the physics literature, Hamiltonian dynamics describe the evolution of (θ,γ) that keeps the total energy H(θ,γ) constant.5 A commonly used technical definition of nonparametric Bayesian models is probability models with infinitely many parameters (Bernardo and Smith, Citation1994).6 All animations were created using the R package animation (Xie, Citation2013) and imported to LATEXusing the package animate (Grahn, Citation2021).7 mlogit does not take into account the time dimension. With our dataset, mlogit failed to converge for alternative-specific price coefficients, and hence, we were only able to obtain output for a common price parameter across all choice alternatives.Additional informationFundingSciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.