A generative theory of similarity

We argue that similarity judgments are inferences about generative processes, and that two objects appear similar when they are likely to have been generated by the same process. We describe a formal model based on this idea and show how featural and spatial models emerge as special cases. We compare our approach to the transformational approach, and present an experiment where our model performs better than a transformational model. Every object is the outcome of a generative process. An animal grows from a fertilized egg into an adult, a city develops from a settlement into a metropolis, and an artifact is assembled from a pile of raw materials according to the plan of its designer. Observations like these motivate the generative approach, which proposes that an object may be understood by thinking about the process that generated it. The promise of the approach is that apparently complex objects may be produced by simple processes, an insight that has proved productive across disciplines including biology [18], physics [21], and architecture [1]. To give two celebrated examples from biology, the shape of a pinecone and the markings on a cheetah’s tail can be generated by remarkably simple processes of growth. These patterns can be characterized much more compactly by describing their causal history than by attempting to describe them directly. Leyton has argued that the generative approach provides a general framework for understanding cognition. Applications of the approach can be found in generative theories of perception [12], memory [12], language [3], categorization [2], and music [11]. This paper offers a generative theory of similarity, a notion often invoked by models of high-level cognition. We argue that two objects are similar to the extent that they seem to have been generated by the same underlying process. The literature on similarity covers settings that extend from the comparison of simple stimuli like tones and colored patches to the comparison of highly-structured objects like narratives. The generative approach is relevant to the entire spectrum of applications, but we are particularly interested in high-level similarity. In particular, we are interested in how similarity judgments draw on intuitive theories, or systems of rich conceptual knowledge [15]. Generative processes and theories are intimately linked. Murphy [14], for example, defines a theory as ‘a set of causal relations that collectively generate or explain the phenomena in a domain.’ We hope that our generative theory provides a framework in which to model how similarity judgments emerge from intuitive theories. We develop a formal theory of similarity and compare it to three existing theories. The featural account [20] suggests that the similarity of two objects is a function of their common and distinctive features, the spatial account suggests that similarity is inversely proportional to distance in a spatial representation, [19] and the transformation account suggests that similarity depends on the number of operations required to transform one object into the other [6]. We show that versions of each of these approaches emerge as special cases of our generative approach, and present an experiment that directly compares our approach with the transformation account. A fourth theory suggests that similarity relies on a process of analogical mapping [5]. We will not discuss this approach in detail, but finish by suggesting how a generative approach to analogy differs from the standard view. Generative processes and similarity Before describing our formal model, we give an informal motivation for a generative approach to similarity. Suppose we are shown a prototype object and asked to describe similar objects we might find in the world. There are two kinds of answers: small perturbations of the prototype, or objects produced by small perturbations of the process that generated the prototype. The second strategy is likely to be more successful than the first, since many perturbations of the prototype will not arise from any plausible generative process, and thus could never appear in practice. By construction, however, an object produced by a perturbation of an existing generative process will have a plausible causal history. To give a concrete example, suppose the prototype is a bug generated by a biological process of growth (Figure 1ii). The bug in i is a small perturbation of the prototype, but seems unlikely to arise since legs are generated in pairs. A perturbation of the generative process might produce a bug with more segments, such as the bug in iii. If we hope to find a bug that is similar but not identical to the prototype, iii is a better bet than i. A sceptic might argue that this one-shot learning problem can be solved by taking the intersection of the set of objects similar to the prototype and the set of obi) ii) Prototype iii) Figure 1: Three bugs. Which is more similar to the prototype — i or iii? jects that are likely to exist. The second set depends critically on generative processes, but the first set (and therefore the notion of similarity) need not. We think it more likely that the notion of similarity is ultimately grounded in the world, and that it evolved for the purpose of comparing real-world objects. If so, then knowledge about what kinds of objects are likely to exist may be deeply bound up with the notion of similarity. The one-shot learning problem is of practical importance, but is not the standard context in which similarity is discussed. More commonly, subjects are shown a pair of objects and asked to rate the similarity of the pair. Note that both objects are observed to exist and the previous argument does not apply. Yet generative processes are still important, since they help pick out the features critical for the similarity comparison. Suppose, for instance, that a forest-dweller discovers a nutritious mushroom. Which is more similar to the mushroom: a mushroom identical except for its size, or a mushroom identical except for its color? Knowing how mushrooms are formed suggests that size is not a key feature. Mushrooms grow from small to large, and the final size of a plant depends on factors like the amount of sunlight it received and the fertility of the soil that it grew in. Reflections like these suggest that the differently-sized mushroom should be judged more similar. A final reason why generative processes matter is that they are deeply related to essentialism. Medin and Ortony [13] note that ‘surface features are frequently constrained by, and sometimes generated by, the deeper, more central parts of objects.’ Even if we observe only the surface features of two objects, it may make sense to judge their similarity by comparing the deeper properties inferred to generate the surface features. Yet we can say more: just as surface features are generated by the essence of the object, the essence itself has a generative history. Surface features are often reliable guides to the essence of an object, but the object’s causal history is a still more reliable indicator, if not a defining criterion of its essence. Keil [9] discusses the case of an animal that is born a skunk, then undergoes surgery that leaves it looking exactly like a raccoon. Since the animal is generated in the same way as a skunk (born of skunk parents), we conclude that it remains a skunk, no matter how it appears on the surface. These examples suggest that the generative approach may help to explain a broad class of theory-dependent inferences. We now present a formal model that attempts to capture the intuitions behind all of these cases. A computational theory of similarity Given a domain D, we develop a theory that specifies the similarity between any two samples from D. A sample from D will usually contain a single object, but working with similarities between sets of objects is useful for some applications. We formalize a generative process as a probability distribution over D that depends on parameter vector θ. Suppose that s1 and s2 are samples from D. We consider two hypotheses: H1 holds that s1 and s2 are independent samples from a single generative process, and H2 holds that the samples are generated from two independently chosen processes. Similarity is defined as the probability that the objects are generated by the same process: that is, the relative posterior probability of H1 compared to H2: sim(s1, s2) = P (H1|s1, s2)