Mathematical Psychology
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Exemplar Models (GCM)

The Generalized Context Model (GCM) proposes that categorization decisions are based on the summed similarity of a new item to all stored exemplars of each category.

P(A|x) = Σsᵢₐ / (Σsᵢₐ + Σsⱼᵇ)

Exemplar models, particularly Robert Nosofsky's Generalized Context Model (GCM, 1986), propose that categorization involves comparing new stimuli to individually stored category exemplars rather than abstract prototypes or rules. The GCM has become one of the most successful and thoroughly tested models in the categorization literature.

The GCM

Generalized Context Model dᵢⱼ = [Σ wₖ |xᵢₖ − xⱼₖ|ʳ]¹ᐟʳ
sᵢⱼ = exp(−c · dᵢⱼ) (exponential similarity)
P(A|x) = βₐ Σ s(x, eₐ) / [βₐ Σ s(x, eₐ) + βᵇ Σ s(x, eᵇ)]

The model computes the psychological distance between items using a weighted Minkowski metric, then converts distances to similarities using an exponential decay function (motivated by Shepard's universal law of generalization). Attention weights (wₖ) can be selectively allocated to relevant dimensions, capturing the finding that people learn to attend to diagnostic dimensions during categorization learning.

Empirical Success

The GCM provides excellent quantitative fits to a wide range of categorization phenomena, including prototype effects (which emerge naturally as a statistical consequence of summing over many exemplars), typicality gradients, the effects of category size, and individual differences in categorization strategy. Its success demonstrated that many phenomena previously attributed to abstraction processes can be explained by exemplar storage and similarity-based retrieval.

Related Topics

References

  1. Nosofsky, R. M. (1986). Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 115(1), 39–57. https://doi.org/10.1037/0096-3445.115.1.39
  2. Medin, D. L., & Schaffer, M. M. (1978). Context theory of classification learning. Psychological Review, 85(3), 207–238. https://doi.org/10.1037/0033-295X.85.3.207
  3. Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323. https://doi.org/10.1126/science.3629243
  4. Kruschke, J. K. (1992). ALCOVE: An exemplar-based connectionist model of category learning. Psychological Review, 99(1), 22–44. https://doi.org/10.1037/0033-295X.99.1.22

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