Foundations of Inference

There are four obstacles to inference which science must overcome.

OBSTACLESSOLUTIONSORIGINATORS
UNCERTAINTY
have => want
now => later
statistic => parameter
PROBABILITY
distribution
regular irregularity
misfit detection
Bernoulli 1713
De Moivre 1733
Laplace 1774
Poisson 1837
DISTORTION
non-linearity
unequal intervals
incommensurability
ADDITIVITY
linearity
arithmetic
concatenation
Luce/Tukey 1964
Fechner 1860
Helmholtz 1887
N.Campbell 1920
CONFUSION
interdependence
interaction
confounding
SEPARABILITY
sufficiency
invariance
conjoint order
Rasch 1960
R.A.Fisher 1920
Thurstone 1925
Guttman 1944
AMBIGUITY
arbitrary grouping
ambiguous hierarchy
DIVISIBILITY
independence
stability
reproducibility
exchangeability
Kolmogorov 1932
Levy 1924
Bookstein 1992
de Finetti 1931

1. Uncertainty is the motivation for inference. We have only the past with which to foresee the future. The past may seem certain. But, on reflection, it is realized to have been not only unique and unrepeatable - awash in special circumstances and inexplicable outcomes - but also a mere interpretation. In order to generalize the problematic past, we invent probability distributions that describe how idiosyncratic data can result from regular processes. It is probability that enables us to make the past useful for predicting the future.

2. Distortion occurs during the transition from observation to conceptualization. Data have no meaning of their own, they must always be interpreted. This interpretation is expedited when we transform our data in ways that facilitate thinking. Our ability to figure things out is rooted in our ability to visualize. Our ability to visualize devolved from the survival value of bodily navigation. Thus, the best solution to distortion is to represent observations in a linear, i.e., additive, form that makes our observations look like the space in front of us. To "see" what our data "mean", we "picture" them.

3. Confusion is caused by the inevitability of inter-action and interdependency. Raw data are too complicated to understand. Each data point indicates many things, intended and unintended. As we look for tomorrow's probabilities in yesterday's lessons, confounding confuses us. Our solution is to force the endless complexity we experience into only a few invented "dimensions", few enough that we can think clearly with them. The authority of these fictions is their logic and utility. Their "truth" is intangible. What matters is whether they work - a consequence discoverable in future experience.

The logic necessary to control confounding is enforced singularity. Define and measure one dimension at a time. The mathematics necessary is parameter separability. Only models which introduce hypothesized "causes" as separately estimable parameters survive confounding. The models which do this are the founding laws of quantitative research. They define measurement. They determine what is measurable. They decide which data are useful.

4. Ambiguity, a different form of confusion, is also present, because we never know precisely what to count or at what level of detail. To measure traffic, do we count cars or passengers? Do we rate performance holistically or itemize? Robust inference requires that incidental changes in data collection cause no change in inference. Ambiguity is controlled by measurement models that possess divisibility and thus are stable with respect to arbitrary composition. Divisibility, the solution to ambiguity, is algebraically equivalent to separability, the solution to confusion.

The Table summarizes the obstacles to inference, their current solutions and the originators of these solutions.

Benjamin D. Wright 1994 RMT 8:1 p. 346


Foundations of inference. Wright BD. … Rasch Measurement Transactions, 1994, 8:1 p.346



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