Structured Probabilistic Models: A New Direction in Cognitive Science J. McClelland SymSys 100 June 1, 2010

The Idea of the Hypothesis Space (Shepard, 1987) Generic question: Suppose item x in the space at right is called ‘a dax’. What about items y and z? People are more likely to think that y is a dax than z. Why? According to Shepard, they implicitly entertain the possibility that the region of the space associated with the label dax is a convex shape, but its size could be anything from very small to very large. This is the hypothesis space x must be in the region in question, so this rules out many of the regions. What about y and z? y is in more of the remaining possible regions than z. x y z

Using evidence to constrain the hypothesis space. x x q q y z Suppose you know that x and y are both daxes. Is q a dax? Suppose you know that x and z are both daxes. Is q a dax? We tend to assume that items are sampled fairly, so the region including x and y is likely to be smaller than the region including x and z; q is more likely to be in the {x,z} region than the {x,y} region.

If the animals on the left are both daxes, what about the animals on the right?

If the animals on the left are both daxes, what about the animals on the right?

If the animals on the left are both daxes, what about the animals on the right?

Xu & Tenenbaum’s Structured Probabilistic Model Subject starts with a structured hypothesis space: the name could be picking out one of several alternative levels within the structure at right. Again the subject is thought to assume that the items are sampled fairly, so that if they both come from the same branch of the tree this increases the probability that the hypothesis is more restricted.

Applications Selection of alternative structure types Complex or simple rules for categorizing stimuli Generative Grammar vs Other Alternatives to capture the structure of natural language Taxonomic Hierarchy vs Two-Dimensional Projection Probabilistic ‘Language of Thought’

The Debate Although supporters of structured probabilisitic models point out that experience (evidence) shapes selection of the alternative chosen, others (like me) think that they are still often building too much in. Also, specific structure types that participants are thought to choose between many be too restrictive No single structure type might actually correctly characterize natural language or the similarity relations among concepts. Typically, the approach ignores process (computations to select among alternatives are complex) But in many studies behavior conforms to the predictions of such models So they are likely to contribute to cognitive science and lead to improved models of cognition in the future.