Abstract
We describe the inductive problem of determining the dimensionality of a latent structure. We show that induction is impossible without some restriction on the form of the output mapping. We argue that monotonicity is the weakest sufficient constraint. First, monotonicity strikes an appropriate balance between generality on the one hand and tractability on the other hand. Second, while it may be desirable to adopt a more restrictive assumption than monotonicity, in most situations this cannot be justified. This problem is not peculiar to psychological science but is a case of the general problem of nomic measurement that affects all branches of science. We illustrate the problem of nomic measurement, and clarify why monotonicity is a reasonable practical solution.
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Notes
- 1.
We exclude for the present the hypothesis that dim(LV space) = 0 where the state-trace consists of a single point in DV space.
- 2.
If x is a linear function of u and y is a linear function of u, then y is a linear function of x and vice versa.
- 3.
That is, defining a testable property of the state-trace that identifies it as an element of Q.
- 4.
This assumes that c ≤ d.
- 5.
As it turns out, mercury and many other substances, including water, do not expand linearly with temperature. Hence the need for careful calibration.
- 6.
Although different interpretations are possible. For example, one latent variable may correspond to memory for mono-oriented objects and the second to the difference between this and memory for faces.
- 7.
This is quantified by a constant known as the thermal expansion coefficient which is about 3.5 times greater for alcohol than for water—which is why alcohol is used in thermometers instead of water.
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Dunn, J.C., Kalish, M.L. (2018). Monotonicity. In: State-Trace Analysis. Computational Approaches to Cognition and Perception. Springer, Cham. https://doi.org/10.1007/978-3-319-73129-2_2
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DOI: https://doi.org/10.1007/978-3-319-73129-2_2
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