The applicability of Bayesian belief networks for measuring user preferences: some numerical simulations

Bayesian belief networks offer an alternative to conventional estimation methods in estimating user preference or utility functions. Because parameter estimates are updated sequentially, this approach seems very promising in user-centred design and data collection systems. The application of such networks however poses several questions, related to speed of learning, sample heterogeneity and discretionalisation of the parameter space. This paper reports the results of a series of numerical simulations which were conducted to gain more insights into these operational decisions. doi:10.1068/b32054 ôCorresponding author. the variables, and (b) a conditional probability table for each node, representing beliefs conditional upon the possible states of parent nodes, if any. A BBN can be used to compute posterior probability distributions representing the updated beliefs of unobserved variables, given (new) evidence for one or more observed variables in the network, through forward and backward reasoning. Since updated beliefs represent the prior probabilities for belief updating when the next evidence is presented, the system is able to learn incrementally. In the present application, the nodes of the network represent users' characteristics, preferences, and design variables, and the links denote the causal relationships between them. Using an existing method of probability propagation in BBNs, the system makes inferences and updates beliefs each time user input is provided. In this application preferences are unobserved variables and choices are observed and causally dependent variables. Recent work in the area of discrete choice modelling has shown that the Bayesian method of estimating parameters from choice observations is asymptotically consistent with log-likelihood estimates in a multinomial logit framework [for an overview of this field of research, see Train (2003)]. In contrast to log-likelihood estimation, the Bayesian method does not make assumptions about the form of the log-likelihood function and, therefore, is more robust than the more conventional estimation methods. There are two additional advantages that are particularly important in the present study. First, Bayesian belief updating is not limited to a single layer of independent variables, but is defined for more complex (causal) networks in general. In the present application this is important, because design choices generally interact in terms of preferences as well as physical constraints. A network is able to represent the complex relationships that cannot be captured in linear (additive or multiplicative) utility functions commonly used in discrete-choice models. Second, Bayesian belief updating supports incremental learningöthat is, learning on a case-by-case basis. Although Bayesian estimation of choice models is now an established method, application of the method in the context of a full network is not straightforward. There are several operational problems that need to be addressed. First, existing algorithms for probability propagation are readily available through software, such as Hugin and Netica (Norsys, 1997), but require that states of variables in the network are discretisised. Belief networks in which continuous and discrete variables are intermixed, are known as hybrid belief networks. The problem of discretisation of continuous variables in such networks has received attention and heuristic methods to find efficient discretisations that minimise information loss have been proposed. For example, Kozlov and Koller (1997) proposed a method that uses relative entropy (or Kullback ^ Leiber distance) as a criterion for partitioning a continuous (density) function. AgenaRisk (2005) is an example of BBN software that supports discretisation methods based on similar principles. These methods are, however, not applicable in the present context as they assume that the amount of information loss for any tentative discretisation can be measured. In the present application the preference probability distributions are not known a priori but need to be learned incrementally from choice data provided on a case-by-case basis, so this assumption does not hold. This means that discretisations of the preference variables need to be predefined. Thus, a first question that needs to be addressed is: How robust is the system for discretising and what magnitude of resolution is required to obtain satisfactory results? Second, learning speed determines the number of cases needed to reach a desired level of knowledge about users preferences. The speed may be dependent on the size of error variance in choice behaviour and preference heterogeneity in the training sample. A next question to be considered, therefore, is: what is the impact of the size of nonsystematic variation in choice behaviour and taste heterogeneity in a group of users on the speed of learning? 2 M A Orzechowski, T A Arentze, AW J Borgers, H J P Timmermans