Nonparametric Bayesian Discrete Latent Variable Models for ...

4.5 A Choice Model with Infinitely Many Latent Features
available to the researcher and one would like to infer them from observed choices.
Although the EBA model can account for many choice scenarios, its usefulness for the
analysis of choice data has been extremely limited by the fact that inference about the
underlying features is very difficult (Tversky and Sattath, 1979; Batsell et al., 2003).
In the following, we present a Bayesian formulation and analysis of the EBA model.
We employ the IBP as a prior over the latent binary features of alternatives in the choice
set and infer the latent features and their weights (importance of each feature to the
choice maker) using MCMC and demonstrate the performance of the proposed model on
a data set studied in psychology. We describe the non-parametric Bayesian formulation
of the EBA model in the next section. The MCMC algorithm used for inference on the
model is presented in Section 4.5.2, followed by experimental results in Section 4.5.3,
and we conclude with a discussion in Section 4.5.4.
4.5.1 Model Specification
The EBA model is defined for choice from a set of several options but for clarity of
presentation we consider only the paired comparison case here which reduces the EBA
model to Restle’s choice model (Restle, 1961; Tversky, 1972). Nevertheless, the inference
techniques we describe below are also valid for the general EBA model.
In a paired comparison experiment there is a set of N options but the subject is
presented only with pairs of options at a time. The task of the subject is to indicate
which of the two options i and j she prefers. In the EBA model, options are described
by K-dimensional binary vector of features z, referred to as aspects. The probability of
choosing option i over option j is given as
pij =
k wkzik(1 − zjk)
k wkzik(1 − zjk) +
k wkzjk(1 − zik)
, (4.63)
where zik denotes the binary value of the kth feature of option i and wk is the positive
weight associated with the kth feature. The greater the weight of a feature, the heavier
its influence on the choice probabilities. The sum
k wkzik(1 − zjk) collects the weights
for all the aspects that option i has but option j does not have. Therefore, the choice
between two alternatives depends only on the features that are not shared between
the two. Equation (4.63) expresses that the EBA model can account for the effects of
similarity on choice. For the above equation, we define the ratio 0
0
= 0.5. That is,
if two alternatives have exactly the same set of features, the choice probability is 0.5.
On the other extreme, if the options are characterized only by unique aspects, i.e. no
option shares any feature with any other option, the BTL model is recovered. Recently,
Ruan et al. (2005) have developed a choice model using a Poisson race model in which
the choice probabilities are mainly determined by the difference in the features of the
alternatives, but the similarities are also allowed to have some effect. This model can
be seen as a generalization of the EBA model.
If one option i has all the features that another alternative j has and more features on
top of these then i will always be preferred over j. This is a reasonable assumption but
in real choice data it can happen that subjects occasionally fail to choose alternative
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