Preference structures and their numerical representations

AbstractThis paper presents a selective survey of numerical representations of preference structures from the perspective of the representational theory of measurement. It reviews historical contributions to ordinal, additive, and expected utility theories, then describes recent contributions in these areas

Coverage, Continuity and Visual Cortical Architecture

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far. We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise. Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure

What is the source of L1 attrition? The effect of recent L1 re-exposure on Spanish speakers under L1 attrition

The recent hypothesis that L1 attrition affects the ability to process interface structures but not knowledge representations (Sorace, 2011) is tested by investigating the effects of recent L1 re-exposure on antecedent preferences for Spanish pronominal subjects, using offline judgements and online eye-tracking measures. Participants included a group of native Spanish speakers experiencing L1 attrition (‘attriters’), a second group of attriters exposed exclusively to Spanish before they were tested (‘re-exposed’), and a control group of Spanish monolinguals. The judgement data shows no significant differences between the groups. Moreover, the monolingual and re-exposed groups are not significantly different from each other in the eye-tracking data. The results of this novel manipulation indicate that attrition effects decrease due to L1 re-exposure, and that bilinguals are sensitive to input changes. Taken together, the findings suggest that attrition affects online sensitivity with interface structures rather than causing a permanent change in speakers’ L1 knowledge representations

Bayesian Decision Theory and Stochastic Independence

As stochastic independence is essential to the mathematical development of probability theory, it seems that any foundational work on probability should be able to account for this property. Bayesian decision theory appears to be wanting in this respect. Savage’s postulates on preferences under uncertainty entail a subjective expected utility representation, and this asserts only the existence and uniqueness of a subjective probability measure, regardless of its properties. What is missing is a preference condition corresponding to stochastic independence. To fill this significant gap, the article axiomatizes Bayesian decision theory afresh and proves several representation theorems in this novel framework