A Mathematical Theory of Arguments for Statistical Evidence

by Monney, P. A.

• ISBN13: 9783790815276
• ISBN10: 3790815276
• Format: Paperback
• Copyright: 2002-12-01
• Publisher: Physica Verlag

Summary

The subject of this book is the reasoning under uncertainty based on statistical evidence. The concepts are developed, explained and illustrated in the context of the mathematical theory of hints, which is a variant of the Dempster-Shafer theory of evidence. In the first two chapters, the theory of generalized functional models for a discrete parameter is developed, which leads to a general notion of weight of evidence. The second part of the book is dedicated to the study of special linear functional models

Table of Contents

The Theory of Generalized Functional Models	1	(39)
The Theory of Hints on Finite Frames	1	(12)
Definition of a Hint and its Associated Functions	2	(3)
Combination of Hints	5	(2)
Mass functions and gem-functions	7	(3)
Dempster Specialization Matrices	10	(1)
A Representation of the Dempster Specialization Matrix	11	(1)
Combining Several Hints	12	(1)
Application	13	(1)
The Combination of Closed Hints	14	(4)
The Theories of Bayes and Fisher	18	(3)
Jessica's Pregnancy Test	18	(1)
The Solution of Bayes	18	(1)
The Solution of Fisher	19	(2)
The Definition and Analysis of a Generalized Functional Model	21	(3)
Generalized Functional Models and Hints	24	(2)
Examples of Generalized Functional Models	26	(7)
Jessica's Pregnancy Test	26	(2)
Policy Identification (I)	28	(3)
Policy Identification (II)	31	(2)
Prior Information	33	(6)
The Plausibility and Likelihood Functions	39	(20)
The Likelihood Ratio as a Weight of Evidence	39	(1)
The Weight of Evidence for Composite Hypotheses	40	(3)
Functional and Distribution Models	43	(6)
The Distribution Model of the Problem	43	(1)
A GFM Obtained by Conditional Embedding	44	(2)
A GFM Inspired by Dempster's Structures of the First kind	46	(3)
Evidence About a Survival Rate	49	(6)
Degrees of Support as Weights of Evidence	55	(4)
Hints on Continuous Frames and Gaussian Linear Systems	59	(12)
Continuous Hints	59	(3)
Gaussian Hints	62	(2)
Precise Gaussian Hints	64	(4)
Gaussian Linear Systems	68	(3)
Assumption-Based Reasoning with Classical Regression Models	71	(26)
Classical Linear Regression Models as Special Gaussian Linear Systems	71	(1)
The Principle of the Inference	72	(4)
The Result of the Inference	76	(3)
Changing Basis	79	(3)
Permissible Bases and Admissible Matrices	82	(4)
Different Representations of the Result of the Inference	86	(4)
A Representation Derived from Linear Dependencies	86	(2)
A Representation Derived from a Special Class of Admissible Matrices	88	(2)
Full Rank Matrices T2 such that T2A = 0	90	(7)
A Matrix Based on Linear Dependencies	90	(1)
A Matrix Based on the Householder Method	91	(1)
A Matrix Based on Classical Variable Elimination	91	(2)
A Matrix Based on a Generalized Inverse of A	93	(4)
Assumption-Based Reasoning with General Gaussian Linear Systems	97	(12)
The Principle of the Inference	97	(1)
The Gaussian Hint Inferred from a Gaussian Linear System of the First Kind	98	(1)
The Gaussian Hint Inferred from a Gaussian Linear System of the Second Kind	99	(5)
The Gaussian Hint Inferred from a Gaussian Linear System of the Third Kind	104	(1)
The Gaussian Hint Inferred from a Gaussian Linear System of the Fourth Kind	105	(1)
Canonical Proper Potentials	106	(3)
Gaussian Hints as a Valuation System	109	(20)
Shenoy-Shafer's Axiomatic Valuation Systems	109	(1)
Marginalization of Gaussian Hints	110	(7)
Marginalization of a Precise Gaussian Hint	111	(1)
Marginalization of a Non-Precise Gaussian Hint	112	(5)
Vacuous Extension of Gaussian Hints	117	(1)
Transport of Gaussian Hints	117	(1)
Projection and Composition of Potentials	118	(3)
Combination of Gaussian Hints Defined on the Same Domain	121	(2)
Computation of Combined Gaussian Hints	123	(4)
Combination when the Composed Potential is of the Thrid Kind	123	(1)
Combination with a Precise Gaussian Hint	124	(1)
Combination of Two Precise Gaussian Hints	125	(1)
Combination when the Composed Potential is of the First Kind	126	(1)
Combination of Hints Defined on Arbitrary Domains	127	(2)
Local Propagation of Gaussian Hints	129	(8)
The Axioms of Shenoy and Shafer	130	(3)
The Local Propagation Algorithm	133	(4)
Application to the Kalman Filter	137	(12)
Definition of the Random System	137	(1)
Derivation of the Kalman Filter	138	(11)
References	149	(4)
Index	153