Exploring probabilistic integration for the estimation of the mixed multinomial logit model

The traditional numerical integration method for estimating the mixed multinomial logit model is Monte Carlo Simulation, either based on pseudo-random or quasi-random sequences. The quasi-random sequence used in this paper is the Halton sequence. In recent years, a different approach to numerical integration, called probabilistic integration, has gained more traction. Probabilistic integration uses the uncertainty inherent to numerical integration caused by the impossibility of being able to evaluate the integrand at an infinite amount of points. This paper seeks to explore the possibility of applying one such method called Bayesian cubature to the estimation of the mixed multinomial logit model, and see if it is a viable method for this estimation process. To illustrate Bayesian cubature, Bayesian Quasi-Monte Carlo with a Gaussian kernel is used. The results of this type of Bayesian cubature do not approach the results obtained by the traditional methods for a mixed logit with a normal distribution. However, more advanced Bayesian cubature methods might yield stronger results, especially when more complex distributions are used for the mixed logit, and drawing many states is computationally intensive.