Miszellen

David Brenner/D.A.S. Fraser/G. Monette

Theories of Inference or Simple Additives

I. Introduction

Statistical inference is the process of passing from an initial given, a statistical model and data jointly called an inference base, to a presentation of information concerning the unknowns. In the literature such information is often re- ferred to as conclusions, partly to emphasize a distinction from the context of decisions. Here the view is taken that such presented information can come only from what is avail- able, the Hiven, and that statistical inference is a deductive process. From this it follows that various theories of in- ference can differ only in the elements that are taken in ad- dition to the initially given inference base. These additives are sometimes recognized and made explicit but often are mere- ly implicit and diffuse perturbing and deflecting the deduc- tive process and ultimately subverting the conclusions. Thus, in any discussion of statistical inference our attention focuses on establishing the given and then determining the additives beyond the inference base.

2. A Deductive Process

A statistical investigation yields two basic entities: The model b% summarizes the'background information as it bears on the variables and unknowns of the investigation; this is a description of the system - w~th variables for components of the system and parameters for unknown characteristics and, going beyond the deterministic case, with probability descrip- tions for the variation or randomness; the data ~ are the values from the investigation for the observable variables of the model. The core view is that statistical inference is the actual process of determining the information contained in the model~, and data ~ concerning the unknowns in the system under investigation; of determining what is directly implied by (~,~) in the ordinary logical and mathematical-deductive sense. Some cases do exist where this process produces exact categorical and frequency (probability) statements for the unknowns. Many others, however, are not so directly amenable and various additional elements intrude explicitly or impli- citly into the deductive process - the additives.

3. Theories of Inference

The terms "theory of inference", "model for inference", "new ways of reasoning towards ..." exist rather commonly in the

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literature. The suggestion is that some process above ordinary logic can somehow produce information beyond that contained in the model and the data. While this near mystical or divine hope may seem appealing, the view here is that one cannot ob- tain more than is contained in the given material, be this the model and data alone or with additives. This focuses directly on the clear expression of the given material for any discus- sions of inference.

4. The Additives

Inference as a deductive process can depend only on the given material. For a specified model and the data, the various theories of inference may then differ only in the additives inserted explicitly or implicitly at various points of the process. The additives vary from formal principles such as sufficiency, ancillarity, or likelihood, to criteria for choosing components for analysis, to larger frames of re- ference. These additives, if viewed as part of the basic given material, may require rephrasing or reexpression in order to be seen as explicit additives at the initial model and data stage. The deductive process is then involved with (Th,~,~) where ~ denotes collectively the additives formally being ap- pended to the inference base (~,~). An examination of the in- ference process thus focuses directly on determining the ad- ditives precisely.

An examination of various considerations of statistical in- ference and of various theories of inference indicates that frequently the additives are not clearly formalized or isolated and may even be difficult to elicit from discussions. If the meaning of conclusions from an inference process is to be clear then the starting point, the given material, must cor- respondingly be clear. Just what the primary additives are then becomes the essential question.

5. Models

The model for an investigation is properly viewed as providing a reasonable approximation to the realities of an investiga- tion; the process of formalizing the model from the background information directly involves this approximation. Accordingly a given investigation may give rise to different approximations as represented in different models, some viewed as closer ap- proximations, some viewed as closer in certain aspects and looser in others.

The preceding in no way lessens the imperative concerning the determination of the additives. In fact, it reinforces the imperative. If various models are being entertained and if additives are involved, then the various conclusions are cor- respondingly qualified by the additives involved and their relative substance and merit is of fundamental concern.

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6. Overview

It is argued that the reference to theories of inference is inappropriate and even misleading. The essential focus is on the determination of the additives that get coupled with the model and data for the deductive process that is statistical inference.

Summary

The view is taken that statistical inference is a deductive process. Theories of inference then differ in the additives that are explicitly or implicitly added to the model and data. This focuses on the clear determination of the given for any inference analysis.

Keywords:

inference, statistical inference, additives, model, data, deductive process.

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