Testing probabilistic models of choice using column generation

Abstract

In so-called random preference models of probabilistic choice, a decision maker chooses according to an unspecified probability distribution over preference states. The most prominent case arises when preference states are linear orders or weak orders of the choice alternatives. The literature has documented that actually evaluating whether decision makers’ observed choices are consistent with such a probabilistic model of choice poses computational difficulties. This severely limits the possible scale of empirical work in behavioral economics and related disciplines. We propose a family of column generation based algorithms for performing such tests. We evaluate our algorithms on various sets of instances. We observe substantial improvements in computation time and conclude that we can efficiently test substantially larger data sets than previously possible.

Multinomial models with linear inequality constraints: Overview and improvements of computational methods for Bayesian inference

“…Moreover, the algorithms could be adapted by taking into account which of the inequality constraints are violated for a given dataset. For instance, if 95% of the constraints are satisfied descriptively, it is likely that most of the posterior samples from the encompassing model will also adhere to these constraints (see Smeulders et al (2018) for applications of similar computational heuristics). To approximate the Bayes factor, it might thus be beneficial to reorder the inequalities in the Ab-representation by their relative strength (i.e., by the rejection probability).…”

Nonparametric Analysis of Random Utility Models: Computational Tools for Statistical Testing

“…1 At this point, we indicate that the computational problems handled in the current paper are similar to those encountered in the study of random utility models in binary choice settings with rational choice types represented by strict linear orders over the choice alternatives (Block and Marschak, 1960). Smeulders et al (2018) propose column generation algorithms for that particular setting.…”

“…More generally, the computational problems handled in the current paper are similar to those encountered in the study of random utility models in binary choice settings with rational choice types represented by strict linear orders over the choice alternatives (Block and Marschak (1960)). Smeulders, Davis‐Stober, Regenwetter, and Spieksma (2018) proposed column generation algorithms for this particular setting. Finally, Strazlecki (2017) provided a recent overview of random utility models with a formal structure that is similar to the one of McFadden and Richter (1990)'s model.…”

Ising formulations of some graph-theoretic problems in psychological research: Models and methods

“…This question is of special interest in order-constrained Bayesian inference where the Bayes factor for a large class of models can be partly defined via the probability of a sample from an unconstrained posterior distribution satisfying a set of order constraints; these order constraints, in turn, can often be described as the convex hull of a set of vertices (Klugkist & Hoijtink, 2007). While this is often an easy computational task, calculating the Bayes factor in this way for very complex models can become computationally challenging, see Smeulders, Davis-Stober, Regenwetter, and Spieksma (2018) for an overview and novel solution using column-generation methods. The second question has more of a psychological interpretation: Given a point inside the convex hull of C, we seek a minimal description of this point as a convex combination of elements of C, i.e., a representation using the fewest elements from C. As we show in our decision-theoretic illustration that follows, this question is tantamount to asking: What is the smallest number of preference states that are needed to represent an individual's pattern of choices and what is the corresponding probability distribution over those preference states?…”