TY - JOUR
T1 - Bayesian generalization with circular consequential regions
AU - Griffiths, Thomas L.
AU - Austerweil, Joseph L.
N1 - Funding Information:
We thank Michael Lee, Dan Navarro, Josh Tenenbaum, Ewart Thomas, and an anonymous reviewer for feedback on a previous draft of this manuscript. This work was supported by grant number FA-9550-10-1-0232 from the Air Force Office of Scientific Research .
PY - 2012/8
Y1 - 2012/8
N2 - Generalization-deciding whether to extend a property from one stimulus to another stimulus-is a fundamental problem faced by cognitive agents in many different settings. Shepard (1987) provided a mathematical analysis of generalization in terms of Bayesian inference over the regions of psychological space that might correspond to a given property. He proved that in the unidimensional case, where regions are intervals of the real line, generalization will be a negatively accelerated function of the distance between stimuli, such as an exponential function. These results have been extended to rectangular consequential regions in multiple dimensions, but not for circular consequential regions, which play an important role in explaining generalization for stimuli that are not represented in terms of separable dimensions. We analyze Bayesian generalization with circular consequential regions, providing bounds on the generalization function and proving that this function is negatively accelerated.
AB - Generalization-deciding whether to extend a property from one stimulus to another stimulus-is a fundamental problem faced by cognitive agents in many different settings. Shepard (1987) provided a mathematical analysis of generalization in terms of Bayesian inference over the regions of psychological space that might correspond to a given property. He proved that in the unidimensional case, where regions are intervals of the real line, generalization will be a negatively accelerated function of the distance between stimuli, such as an exponential function. These results have been extended to rectangular consequential regions in multiple dimensions, but not for circular consequential regions, which play an important role in explaining generalization for stimuli that are not represented in terms of separable dimensions. We analyze Bayesian generalization with circular consequential regions, providing bounds on the generalization function and proving that this function is negatively accelerated.
KW - Bayesian inference
KW - Generalization
KW - Rational analysis
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U2 - 10.1016/j.jmp.2012.07.002
DO - 10.1016/j.jmp.2012.07.002
M3 - Article
AN - SCOPUS:84866142258
SN - 0022-2496
VL - 56
SP - 281
EP - 285
JO - Journal of Mathematical Psychology
JF - Journal of Mathematical Psychology
IS - 4
ER -