Walley P. Statistical Reasoning with Imprecise Probabilities

Chapman and Hall, 1991. — 362 p.As the title indicates, the book is about methods of reasoning and statistical inference using imprecise probabilities. The methods are based on a behavioural interpretation of probability and principles of coherence. The idea for such a book originated in 1982, after I had written two long reports on the mathematics and elicitation of upper and lower probabilities. My experience in teaching and applying the existing theories of statistical inference had convinced me that each of them was inadequate. The Bayesian theory is inadequate, despite its great virtues of coherence, because it requires all probability assessments to be precise yet gives little guidance on how to make them. It seemed natural to investigate whether the Bayesian theory could be modified by admitting imprecise probabilities as models for partial ignorance. Is it possible to reconcile imprecision with coherence, vagueness with rationality? Fortunately the answer is yes!
The basic ideas of the book appeared in the two technical reports (1981, 1982). The coherence principles for conditional probabilities and statistical models were worked out in New Zealand, in 1983. A draft of the book was written in 1984-85, under the title 'Rationality and vagueness', but at that point I decided that the results were interesting enough to be given a more careful examination. That has taken me four years. The result is that I say much more than I originally intended about other ways of modelling ignorance and uncertainty, and about issues such as finite versus countable additivity, conditioning on continuous variables, the likelihood principle, and the incoherence of standard statistical methods. To keep the book to a manageable length, I have omitted a great deal of material from earlier versions, particularly concerning the aggregation of beliefs, decision making with imprecise utilities, and stronger properties of coherence.
The mathematical theory in the book is quite general. The theory could have been presented at a more elementary level, by restricting attention to upper and lower probabilities, or to finite spaces and finitely generated models (as in Chapter 4), but in order to build a useful theory of statistical inference it does seem necessary to consider infinite spaces. (The general theory makes use of methods and results from linear functional analysis, whereas the methods of linear programming can be used in the special case of finitely generated models.) The mathematical theory is, I believe, quite simple when one becomes familiar with the basic properties of upper and lower previsions.
The book emphasizes the foundations, principles and rules of probabilistic reasoning, rather than their practical implementation in specific applications. I was tempted to include some more substantial applications of the theory to real problems, as in Walley and Campello de Souza (1986), especially to illustrate how imprecise probabilities can be assessed in practice. However, the complexities of any practical problem demand special assessment strategies and probability models, and an adequate description of these would have added substantially more to an already long manuscript. Readers who are interested in assessment methods and practical implementation could begin by reading Chapter 4, which is concerned with finite models rather than the general theory. Those who want to see specific statistical models should refer to sections 5.3, 5.4, 7.8 and 9.6.Reasoning and behaviour
Coherent previsions
Extensions, envelopes and decisions
Assessment and elicitation
The importance of imprecision
Conditional previsions
Coherent statistical models
Statistical reasoning
Structural judgementsAppendices
A. Verifying coherence
B. n-Coherence
C. Win-and-place betting on horses
D. Topological structure of L and P
E. Separating hyperplane theorems
F. Desirability
G. Upper and lower variances
H. Operational measurement procedures
I. The World Cup football experiment
J. Regular extension
K. W-coherence