Generative Inferences Based on a Discriminative Bayesian Model of Relation Learning

Generative Inferences Based on a Discriminative Bayesian Model of Relation Learning Dawn Chen 1 (sdchen@ucla.edu) Hongjing Lu 1, 2 (hongjing@ucla.edu) Keith J. Holyoak 1 (holyoak@lifesci.ucla.edu) Departments of Psychology 1 and Statistics 2 University of California, Los Angeles Los Angeles, CA 90095 USA formal grammar or a set of logical rules that generates alternative relational “theories”, which are in turn used to predict the observed data. That is, the set of possible relational structures is provided to the system by specifying a grammar that generates them. Despite their impressive successes, there are some reasons to doubt whether the generative approach provides an adequate basis for all psychological models of relation learning. Since the postulated grammar of relations is not itself learned, the generative approach implicitly makes rather strong nativist assumptions. Moreover, generative models of relation learning do not fit the intuitive causal direction. For example, it seems odd to claim that a binary relation such as larger than somehow acts to causally generate an ordered pair (e.g., ) that constitutes an instantiation of the relation. It seems more natural to consider how observable features of the objects in the ordered pair give rise to the truth of the relation, i.e., to apply a discriminative approach. Abstract Bayesian Analogy with Relational Transformations (BART) is a discriminative model that can learn comparative relations from non-relational inputs (Lu, Chen & Holyoak, 2012). Here we show that BART can be extended to solve inference problems that require generation (rather than classification) of relation instances. BART can use its generative capacity to perform hypothetical reasoning, enabling it to make quasi- deductive transitive inferences (e.g., “If A is larger than B, and B is larger than C, is A larger than C?”). The extended model can also generate human-like instantiations of a learned relation (e.g., answering the question, “What is an animal that is smaller than a dog?”). These modeling results suggest that discriminative models, which take a primarily bottom-up approach to relation learning, are potentially capable of using their learned representations to make generative inferences. Keywords: Bayesian models; generative models; discriminative models; relation learning; transitive inference; deduction; induction; hypothetical reasoning Introduction Discriminative Models of Relation Learning Generative and Discriminative Models Bayesian models of inductive learning can be designed to focus on learning either the probabilities of observable features given concepts (generative models) or the probabilities of concepts given features (discriminative models; Friston et al., 2008; Mackay, 2003). Generative models are especially powerful as they are capable of not only classifying novel instances of concepts (using Bayes’ rule to invert conditional probabilities), but also generating representations of possible instances. In contrast, discriminative models focus directly on classification tasks, but do not provide any obvious mechanism for making generative inferences. In recent years, generative Bayesian models have been developed to learn complex concepts based on relational structures (e.g., Goodman, Ullman & Tenenbaum, 2011; Kemp & Jern, 2009; Kemp, Perfors & Tenenbaum, 2007; Tenenbaum, Kemp, Griffiths & Goodman, 2011). Representations of alternative relational structures are used to predict incoming data, and the data in turn are used to revise probability distributions over alternative structures. The highest level of the structure typically consists of a Recently, discriminative models have also been applied to relation learning. Silva, Heller, and Ghahramani (2007) developed a discriminative model for relational tasks such as identifying classes of hyperlinks between webpages and classifying relations based on protein interactions. Although their model was developed to address applications in machine learning, the general principles can potentially be incorporated into models of human relational learning. One key idea is that an n-ary relation can be represented as a function that takes ordered sets of n objects as its input and outputs the probability that these objects instantiate the relation. The model learns a representation of the relation from labeled examples, and then applies the learned representation to classify novel examples. A second key idea is that relation learning can be facilitated by incorporating empirical priors, which are derived using some simpler learning task that can serve as a precursor to the relation learning task. These ideas were incorporated into Bayesian Analogy with Relational Transformations (BART), a discriminative model that can learn comparative relations from non- relational inputs (Lu, Chen & Holyoak, 2012). Given