Logical inference for inverse problems

Estimating a deterministic single value for model parameters when reconstructing the system response has a limited meaning if one considers that the model used to predict its behaviour is just an idealization of reality, and furthermore, the existence of measurements errors. To provide a reliable answer, probabilistic instead of deterministic values should be provided, which carry information about the degree of uncertainty or plausibility of those model parameters providing one or more observations of the system response. This is widely-known as the Bayesian inverse problem, which has been covered in the literature from different perspectives, depending on the interpretation or the meaning assigned to the probability. In this paper, we revise two main approaches: the one that uses probability as logic, and an alternative one that interprets it as information content. The contribution of this paper is to provide an unifying formulation from which both approaches stem as interpretations, and which is more general in the sense of requiring fewer axioms, at the time the formulation and computation is simplified by dropping some constants. An extension to the problem of model class selection is derived, which is particularly simple under the proposed framework. A numerical example is finally given to illustrate the utility and effectiveness of the method.

[1]  Desmond L. Bell,et al.  The emergence of probability: A philosophical study of early ideas about probability, induction and statistical inference , 1987 .

[2]  M. Kendall,et al.  The Logic of Scientific Discovery. , 1959 .

[3]  Henry Ely Kyburg,et al.  Probability and Inductive Logic , 1970 .

[4]  Gary James Jason,et al.  The Logic of Scientific Discovery , 1988 .

[5]  R. T. Cox Probability, frequency and reasonable expectation , 1990 .

[6]  J. Beck,et al.  Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation , 2002 .

[7]  H. Jeffreys A Treatise on Probability , 1922, Nature.

[8]  F. Ramsey The Foundations of Mathematics and Other Logical Essays , 2001 .

[9]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[10]  E. Jaynes Information Theory and Statistical Mechanics , 1957 .

[11]  J. Beck,et al.  Model Selection using Response Measurements: Bayesian Probabilistic Approach , 2004 .

[12]  M. Lawera Predictive inference : an introduction , 1995 .

[13]  E. S. Pearson,et al.  ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE PART I , 1928 .

[14]  James H. Fetzer Scientific Knowledge: Causation, Explanation, and Corroboration , 1981 .

[15]  B. D. Finetti,et al.  Foresight: Its Logical Laws, Its Subjective Sources , 1992 .

[16]  Paul Humphreys,et al.  Why Propensities Cannot Be Probabilities , 1985, Philosophical Papers.

[17]  John W. Yolton,et al.  THE EMERGENCE OF PROBABILITY: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference , 1976 .

[18]  C. Allen,et al.  Stanford Encyclopedia of Philosophy , 2011 .

[19]  R. T. Cox,et al.  The Algebra of Probable Inference , 1962 .

[20]  Rudolf Carnap,et al.  Philosophy and Logical Syntax , 1979 .

[21]  M. S. Bartlett,et al.  Statistical methods and scientific inference. , 1957 .

[22]  L. M. M.-T. Theory of Probability , 1929, Nature.

[23]  A. Kolmogorov Three approaches to the quantitative definition of information , 1968 .

[24]  J. Beck Bayesian system identification based on probability logic , 2010 .

[25]  J. Schreiber Foundations Of Statistics , 2016 .

[26]  Armin W. Schulz,et al.  Interpretations of probability , 2003 .

[27]  Seymour Geisser,et al.  8. Predictive Inference: An Introduction , 1995 .

[28]  R. A. Fisher,et al.  Statistical methods and scientific inference. , 1957 .