# Foundations of Inference

There are four obstacles to inference which science must overcome.

 OBSTACLES SOLUTIONS ORIGINATORS 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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