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Can the Maximum Entropy Principle Be Explained as a Consistency Requirement?
, 1997
"... The principle of maximumentropy is a general method to assign values to probability distributions on the basis of partial information. This principle, introduced by Jaynes in 1957, forms an extension of the classical principle of insufficient reason. It has been further generalized, both in mathe ..."
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The principle of maximumentropy is a general method to assign values to probability distributions on the basis of partial information. This principle, introduced by Jaynes in 1957, forms an extension of the classical principle of insufficient reason. It has been further generalized, both in mathematical formulation and in intended scope, into the principle of maximum relative entropy or of minimum information. It has been claimed that these principles are singled out as unique methods of statistical inference that agree with certain compelling consistency requirements. This paper reviews these consistency arguments and the surrounding controversy. It is shown that the uniqueness proofs are flawed, or rest on unreasonably strong assumptions. A more general class of 1 inference rules, maximizing the socalled R'enyi entropies, is exhibited which also fulfill the reasonable part of the consistency assumptions. 1 Introduction In any application of probability theory to the pro...
Application of Bayesian inference to fMRI data analysis
 IEEE Transactions on Medical Imaging
, 1999
"... Abstract—The methods of Bayesian statistics are applied to the analysis of fMRI data. Three specific models are examined. The first is the familiar linear model with white Gaussian noise. In this section, the Jeffreys ’ Rule for noninformative prior distributions is stated and it is shown how the po ..."
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Abstract—The methods of Bayesian statistics are applied to the analysis of fMRI data. Three specific models are examined. The first is the familiar linear model with white Gaussian noise. In this section, the Jeffreys ’ Rule for noninformative prior distributions is stated and it is shown how the posterior distribution may be used to infer activation in individual pixels. Next, linear timeinvariant (LTI) systems are introduced as an example of statistical models with nonlinear parameters. It is shown that the Bayesian approach can lead to quite complex bimodal distributions of the parameters when the specific case of a delta function response with a spatially varying delay is analyzed. Finally, a linear model with autoregressive noise is discussed as an alternative to that with uncorrelated white Gaussian noise. The analysis isolates those pixels that have significant temporal correlation under the model. It is shown that the number of pixels that have a significantly large autoregression parameter is dependent on the terms used to account for confounding effects. Index Terms — Autoregressive modeling, Bayesian statistics, functional MRI data analysis, linear timeinvariant systems.
The Promise of Bayesian Inference for Astrophysics
, 1992
"... . The `frequentist' approach to statistics, currently dominating statistical practice in astrophysics, is compared to the historically older Bayesian approach, which is now growing in popularity in other scientific disciplines, and which provides unique, optimal solutions to wellposed problems. The ..."
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Cited by 15 (0 self)
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. The `frequentist' approach to statistics, currently dominating statistical practice in astrophysics, is compared to the historically older Bayesian approach, which is now growing in popularity in other scientific disciplines, and which provides unique, optimal solutions to wellposed problems. The two approaches address the same questions with very different calculations, but in simple cases often give the same final results, confusing the issue of whether one is superior to the other. Here frequentist and Bayesian methods are applied to problems where such a mathematical coincidence does not occur, allowing assessment of their relative merits based on their performance, rather than on philosophical argument. Emphasis is placed on a key distinction between the two approaches: Bayesian methods, based on comparisons among alternative hypotheses using the single observed data set, consider averages over hypotheses; frequentist methods, in contrast, average over hypothetical alternative...
Prior experiences and perceived efficacy influence 3yearolds’ imitation. Dev. Psychol. 44, 275–285. Conflict of Interest Statement: The authors declare that the research was conducted in the absence of any commercial or financial relationships that coul
, 2008
"... Children are selective and flexible imitators. They combine their own prior experiences and the perceived causal efficacy of the model to determine whether and what to imitate. In Experiment 1, children were randomly assigned to have either a difficult or an easy experience achieving a goal. They th ..."
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Cited by 14 (6 self)
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Children are selective and flexible imitators. They combine their own prior experiences and the perceived causal efficacy of the model to determine whether and what to imitate. In Experiment 1, children were randomly assigned to have either a difficult or an easy experience achieving a goal. They then saw an adult use novel means to achieve the goal. Children with a difficult prior experience were more likely to imitate the adult’s precise means. Experiment 2 showed further selectivity—children preferentially imitated causally efficacious versus nonefficacious acts. In Experiment 3, even after an easy prior experience led children to think their own means would be effective, they still encoded the novel means performed by the model. When a subsequent manipulation rendered the children’s means ineffective, children recalled and imitated the model’s means. The research shows that children integrate information from their own prior interventions and their observations of others to guide their imitation.
Maximum entropy reconstruction using derivative information part 2: . . .
, 1995
"... Maximum entropy density estimation, a technique for reconstructing an unknown density function on the basis of certain measurements, has applications in various areas of applied physical sciences and engineering. Here we present numerical results for the maximum entropy inversion program based on a ..."
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Maximum entropy density estimation, a technique for reconstructing an unknown density function on the basis of certain measurements, has applications in various areas of applied physical sciences and engineering. Here we present numerical results for the maximum entropy inversion program based on a new class of information measures which are designed to control derivative values of the unknown densities.
On Generalized Entropies and ScaleSpace
, 1997
"... this paper we show that the generalized entropies are such functionals. It should be noted that this behavior is not seen for the number of critical points: Although critical points most often disappear when scale is increased, creation of critical points with increasing scale is a generic event [16 ..."
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Cited by 13 (3 self)
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this paper we show that the generalized entropies are such functionals. It should be noted that this behavior is not seen for the number of critical points: Although critical points most often disappear when scale is increased, creation of critical points with increasing scale is a generic event [16, 14, 7]. Secondly, generalized entropy is the basis for the theory of MultiFractal [11, 18] and it is known that there are very strong algebraic similarities to the fundamental equations of Statistical Mechanics. These are thus well known functions, and while images are not physical systems in classical thermodynamic sense, Linear ScaleSpace is governed by the Linear Heat Diffusion Equation, and one could thus without great difficulty extend the view of images to be a classical thermodynamical system for which the Linear Heat Diffusion is valid. Such a system is an ideal gas. These interpretations of images will be discussed in detail in this chapter. Finally, as will be demonstrated the generalized entropies offer practical, mathematical well founded functions to study scaling behaviors of images for scaleselection and texture analysis. Related to this work is Vehel et al. [29], where images are studied in the multifractal setting, focusing on certain dimensions, and Brink & Pendock [6], and Brink [5] have used the entropy and the closely related Kullback measure to do local thresholding of images. This article is organized as follows. First, in Section 2 will be given a brief introduction to Linear ScaleSpace and linear entropy. Then, in Section 3 will we discuss the generalized entropies, what the difference is to linear entropy, and what their properties are in ScaleSpace. Following this, in Section 4 we will discuss a physical interpretation of images both from the...
An Introduction to Parameter Estimation Using Bayesian Probability Theory
, 1990
"... . Bayesian probability theory does not define a probability as a frequency of occurrence; rather it defines it as a reasonable degree of belief. Because it does not define a probability as a frequency of occurrence, it is possible to assign probabilities to propositions such as "The probability that ..."
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. Bayesian probability theory does not define a probability as a frequency of occurrence; rather it defines it as a reasonable degree of belief. Because it does not define a probability as a frequency of occurrence, it is possible to assign probabilities to propositions such as "The probability that the frequency had value ! when the data were taken," or "The probability that hypothesis x is a better description of the data than hypothesis y." Problems of the first type are parameter estimation problems, they implicitly assume the correct model. Problems of the second type are more general, they are model selections problems and do not assume the model. Both types of problems are straight forward applications of the rules of Bayesian probability theory. This paper is a tutorial on parameter estimation. The basic rules for manipulating and assigning probabilities are given and an example, the estimation of a single stationary sinusoidal frequency, is worked in detail. This example is su...
The Gibbs Paradox
, 1996
"... : We point out that an early work of J. Willard Gibbs (1875) contains a correct analysis of the "Gibbs Paradox" about entropy of mixing, free of any elements of mystery and directly connected to experimental facts. However, it appears that this has been lost for 100 years, due to some obscurities in ..."
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: We point out that an early work of J. Willard Gibbs (1875) contains a correct analysis of the "Gibbs Paradox" about entropy of mixing, free of any elements of mystery and directly connected to experimental facts. However, it appears that this has been lost for 100 years, due to some obscurities in Gibbs' style of writing and his failure to include this explanation in his later Statistical Mechanics. This "new" understanding is not only of historical and pedagogical interest; it gives both classical and quantum statistical mechanics a different status than that presented in our textbooks, with implications for current research. CONTENTS 1. INTRODUCTION 2 2. THE PROBLEM 3 3. THE EXPLANATION 4 4. DISCUSSION 6 5. THE GAS MIXING SCENARIO REVISITED 7 6. SECOND LAW TRICKERY 9 7. THE PAULI ANALYSIS 10 8. WOULD GIBBS HAVE ACCEPTED IT? 11 9. GIBBS' STATISTICAL MECHANICS 13 10. SUMMARY AND UNFINISHED BUSINESS 17 11. REFERENCES 18 y In Maximum Entropy and Bayesian Methods, C. R. Smith, G. J. E...
Prior Information and Uncertainty in Inverse Problems
, 2001
"... Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate dataindependent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may ..."
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Cited by 12 (5 self)
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Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate dataindependent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may significantly influence the results of inverse calculations. The specification of prior information is especially controversial. How does one quantify information? What does it mean to know something about a parameter a priori? In this tutorial we discuss Bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations. In particular we show that apparently conservative Bayesian choices, such as representing interval constraints by uniform probabilities (as is commonly done when using genetic algorithms, for example) may lead to artificially small uncertainties. We also describe tools from statistical decision theory that can be used to...
The Constraint Rule of the Maximum Entropy Principle
, 1995
"... The principle of maximum entropy is a method for assigning values to probability distributions on the basis of partial information. In usual formulations of this and related methods of inference one assumes that this partial information takes the form of a constraint on allowed probability distri ..."
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The principle of maximum entropy is a method for assigning values to probability distributions on the basis of partial information. In usual formulations of this and related methods of inference one assumes that this partial information takes the form of a constraint on allowed probability distributions. In practical applications, however, the information consists of empirical data. A constraint rule is then employed to construct constraints on probability distributions out of these data. Usually one adopts the rule to equate the expectation values of certain functions with their empirical averages. There are, however, various other ways in which one can construct constraints from empirical data, which makes the maximum entropy principle lead to very different probability assignments. This paper shows that an argument by Jaynes to justify the usual constraint rule is unsatisfactory and investigates several alternative choices. The choice of a constraint rule is also show...