This paper intends to show how the fabled violation of Bell's inequality by the probabilistic spe... more This paper intends to show how the fabled violation of Bell's inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein's principle of local realism, the inequality pertains to a li
I provide a critical reassessment of David Mermin's influential and misleading parable, "Quantum ... more I provide a critical reassessment of David Mermin's influential and misleading parable, "Quantum Mysteries for Anyone", identifying its errors and resolving them with a complete analysis of the quantum experiment it is meant to portray. Accessible to popular readership and requiring no knowledge of quantum physics at all, his exposition describes the curious behaviour of a machine that is designed to parody the empirical results of quantum experiments monitoring the spins of a pair of electrons under various conditions. The mysteries are said to unfold from contradictory results produced by a signal process that is proposed to explain them. I find that these results derive from a mathematical error of neglect, coupled with a confusion of two distinct types of experiments under consideration. One of these, a gedankenexperiment, provides the context in which the fabled defiance of Bell's inequality is thought to emerge. The errors are corrected by the recognition of functional relations embedded within the experimental conditions that have been long unnoticed. A Monte Carlo simulation of results in accord with the actual abstemious claims of quantum theory supports probability values that Mermin decries as unwarranted. However, the distribution it suggests is not definitive, in accord with the expressed agnostic position of quantum theory regarding measurements that cannot be executed. Bounding quantum probabilities are computed for the results of the gedankenexperiment relevant to Bell's inequality which inspired the parable. The problem is embedded in a 3 × 3 design of Stern-Gerlach magnet orientations at two observation stations. Computational resolution on the basis of Bruno de Finetti's fundamental theorem of probability requires the evaluation of a battery of three paired linear programming problems. Though technicalities are ornate, the message is clear. There are no mysteries of quantum mechanics that derive from mistaken understandings of Bell's inequality… for anyone.
Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, 1995
ABSTRACT Our focus is on the structure of conditional quantities and conditional prevision assert... more ABSTRACT Our focus is on the structure of conditional quantities and conditional prevision assertions which are crucial to implementing the fundamental theorem of prevision as an “inference engine”, foreseen in principle and constructed in simple forms for more than a century. I argue that conditional quantities do entail a minimal logical structure as required by the principle of coherency. But attempts to identify a more extensive structure of a logic of conditional events via many-valued logical functions are misdirected. The equivalent roles of arithmetic and of many-valued logics in generating a function space of object quantities are made apparent.
This article displays an application of the statistical method moti- vated by Bruno de Finetti... more This article displays an application of the statistical method moti- vated by Bruno de Finetti's operational subjective theory of probability. We use exchangeable forecasting distributions based on mixtures of linear com- binations of exponential power (EP) distributions to forecast the sequence of daily rates of return from the Dow-Jones index of stock prices over a 20 year period. The operational subjective statistical method for comparing distributions is quite different from that commonly used in data analysis, because it rejects the basic tenets underlying the practice of hypothesis test- ing. In its place, proper scoring rules for forecast distributions are used to assess the values of various forecasting strategies. Using a logarithmic scoring rule, we find that a mixture linear combination of EP distributions scores markedly better than does a simple mixture over the EP family, which scores much better than does a simple Normal mixture. Surprisingly, a mix- ture over a l...
A predictive distribution over a sequence of N + 1 events is said to be "frequency mimicking" whe... more A predictive distribution over a sequence of N + 1 events is said to be "frequency mimicking" whenever the probability for the final event conditioned on the outcome of the first N events equals the relative frequency of successes among them. Infinitely extendible exchangeable distributions that universally inhere this property are known to have several annoying concomitant properties. We motivate frequency mimicking assertions over a limited subdomain in practical problems of finite inference, and we identify their computable coherent implications. We provide some computed examples using reference distributions, and we introduce computational software to generate any specification. The software derives from an inversion of the finite form of the exchangeability representation theorem. Three new theorems delineate the extent of the usefulness of such distributions, and we show why it may not be appropriate to extend the frequency mimicking assertions for a specified value of N to any arbitrary larger size of N. The constructive results identify the source and structure of "adherent masses" in the limit of a sequence of finitely additive distributions. Appendices develop a novel geometrical representation of conditional probabilities which illuminate the analysis.
I reassess the gedankenexperiment of Greenberger, Horne, Shimony, and Zeilinger after twenty-five... more I reassess the gedankenexperiment of Greenberger, Horne, Shimony, and Zeilinger after twenty-five years, finding their influential claim to the discovery of an inconsistency inherent in high dimensional formulations of local realism to arise from a fundamental error of logic. They manage this by presuming contradictory premises: that a specific linear combination of four angles involved in their proposed parallel experiments on two pairs of electrons equals both π and 0 at the same time. Ignoring this while presuming the contradictory implications of these two conditions, they introduce the contradiction themselves. The notation they use in their “derivation” is not sufficiently ornate to represent the entanglement in the double electron spin pair problem they design, confounding their error. The situation they propose actually motivates only an understanding of the full array of symmetries involved in their problem. In tandem with the error now recognised in the supposed defiance o...
The refinement axiom for entropy has been provocative in providing foundations of information the... more The refinement axiom for entropy has been provocative in providing foundations of information theory, recognised as thoughtworthy in the writings of both Shannon and Jaynes. A resolution to their concerns has been provided recently by the discovery that the entropy measure of a probability distribution has a dual measure, a complementary companion designated as “extropy”. We report here the main results that identify this fact, specifying the dual equations and exhibiting some of their structure. The duality extends beyond a simple assessment of entropy, to the formulation of relative entropy and the Kullback symmetric distance between two forecasting distributions. This is defined by the sum of a pair of directed divergences. Examining the defining equation, we notice that this symmetric measure can be generated by two other explicable pairs of functions as well, neither of which is a Bregman divergence. The Kullback information complex is constituted by the symmetric measure of en...
ABSTRACT We study de Finetti’s “fundamental theorem of probability”, reformulated in the finite c... more ABSTRACT We study de Finetti’s “fundamental theorem of probability”, reformulated in the finite case as a computable linear programming problem. The theorem is substantially extended, and shown to have important implications for the subjective theory of statistical inference. It supports an operational meaning for the partial assertion of prevision via asserted bounds. We extend the theorem to apply to quantities more general than events, to allow bounds and orderings on previsions as the input to the programming problem, and even to allow bounds on conditional previsions as the input and output. In a philosophical discussion, prevision is explicitly recognized as a completion of the notion of logical assertion, introduced by Frege. Partial prevision assertions then allow the representation of weaker forms of uncertain knowledge, spanning a range of possible numerical values for a quantity between the assertion of a precise prevision value and the mere unassertive contemplation of the quantity.
All statistical measurements which represent the values of useful unknown quantities have a realm... more All statistical measurements which represent the values of useful unknown quantities have a realm that is both finite and discrete. Thus our uncertainties about any measurement can be represented by discrete probability mass functions. Nonetheless, common statistical practice treats probability distributions as representable by continuous densities or mixture densities. Many statistical problems involve the analysis of sequences of observations that the researcher regards exchangeably. Often we wish to find a joint probability mass function over X 1 , X 2 ,. .. , X n , with interim interest in the sequence of updated probability mass functions f (x i+1 | X i = x i) for i = 1, 2,. .. , n − 1. We investigate how well continuous conjugate theory can approximate real discrete mass functions in various measurement settings. Interest centres on approximating digital Normal mass functions and digital parametric mixtures with continuous Mixture Normal and Normal-Gamma Mixture Normal distributions for such items as E(X i+1 | X i = x i) and V (X i+1 | X i = x i). Digital mass functions are generated by specifying a finite realm of measurements for a quantity of interest, finding a density value of some specified function at each point, and then normalising the densities over the realm to generate mass values. Both a digitised prior mixing mass function and digitised information transfer function are generated and used, via Bayes' Theorem, to compute posterior mass functions. Approximating posterior densities using continuous conjugate theory are evaluated, and the two sets of results compared.
Mathematical models for handling partial knowledge in artificial intelligence, 1995
The operational subjective theory of probability and statistics recognizes Bruno de Finetti’s fun... more The operational subjective theory of probability and statistics recognizes Bruno de Finetti’s fundamental theorem of prevision as providing the basis for any coherent expert system of inferences from assertions of partial knowledge. One session of our workshop was devoted to understanding the formulation and computational structure of this theorem, as explained in the articles1 by Lad, Dickey, and Rahman (1990, 1992). In a word, the fundamental theorem of prevision identifies numerically the coherent implications of any partial prevision assertions (including interval probabilities) for any other cohering prevision or conditional prevision whatsoever. The present paper was discussed at a second session, and presumes the reader’s familiarity with these expositions. For the purpose of self-containment here, a brief summary of the definitions, notation, and theorems that lead to the fundamental theorem of prevision in a finite, discrete context, and a concise statement of the theorem in both its parts are appended to the end of this article. They are excerpted from the forthcoming text by Lad (1994). Included in the excerpt are definitions and a basic theorem pertaining to the judgment to regard a sequence of quantities exchangeably, which are also relevant to developments in the present article.
Applied Bayesian Statistical Studies in Biology and Medicine, 2004
Attending the Asti summer school where the papers in this book were presented were some thirty gr... more Attending the Asti summer school where the papers in this book were presented were some thirty graduate students writing empirical theses using applied statistics. Their topics ranged from the monitoring of commercial fishing stock around Sardinia, to learning about the situation of current Nigerian immigrants to Italy, and to comparing the nutrition and weight gain of children of Chinese immigrants in Italy with children of their non-emigrating cohort in a Chinese village. The thesis students were attending the course to improve their practical statistical skills. Having been involved in the modern development of subjective Bayesian methods for some thirty years myself, I was somewhat surprised by how little was known of these major developments which have transformed statistical thinking throughout the world over this period. They had received major impetus from the research of the noted Italian probabilist, Bruno de Finetti, and his followers.
We consider how to calculate the probability that the sum of the product of variables assessed wi... more We consider how to calculate the probability that the sum of the product of variables assessed with a Normal distribution is negative. The analysis is motivated by a specific problem in electrical engineering. To resolve the problem, two distinct steps are required. First, we consider ways in which we can assess the distribution for the product of two Normally distributed variables. Three different methods are compared: a numerical methods approximation, which involves implementing a numerical integration procedure on MATLAB, a Monte Carlo construction and an approximation to the analytic result using the Normal distribution. The second step considers how to assess the distribution for the sum of the products of two Normally distributed variables by applying the Convolution Formula. Finally, the two steps are combined to compute the distribution for the sum of products of Normally distributed variables, thus to calculate the probability that this sum of products is negative. The pro...
We re-examine the procedure of median decision making in the context of radiological determi- nat... more We re-examine the procedure of median decision making in the context of radiological determi- nation of asbestosis by three B-readers. Our assessment addresses the specificity, sensitivity and predictive values from this procedure compared to merely an individual radiologist's diagnosis. Con- ditional exchangeability of the radiologists' classifications is recognised as more appropriate than independence which is often presumed. The framework of de Finetti's fundamental theorem of prob- ability makes the analysis tractable when it is formulated in terms of a linear programming problem, yielding coherent bounds on probabilities of interest even when a complete distribution is not spec- ified. Further natural assertions motivate a partial ordering of conditional probabilities. In this context the computation of bounds develops into a quadratic programming problem. Using sensible assertions, the median decision procedure is found to be relatively weaker than has been though...
We present the technical details of the logitnormal distribution of a variable supported on the o... more We present the technical details of the logitnormal distribution of a variable supported on the open unit-interval. Although this distribution has been used widely in a variety of sta-tistical applications under a transformation, its direct analysis has not been exposited. The density function can be expressed algebraically, and the moments can be computed easily by numerical integration as functions of the parameters of the transformed normal distribution. The family of densities has characteristics different from the Beta family especially near 0 and 1, although some members of each family resemble each other. Limiting members of the family provide examples of distributions with adherent masses at 0 and 1. We also examine the bivariate exchangeable logitnormal distribution.
This article provides a completion to theories of information based on entropy, resolving a longs... more This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback-Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the L 2 metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.
Annals of Mathematics and Artificial Intelligence, 2002
This paper displays the productive role of the judgment of exchangeability and even conditional e... more This paper displays the productive role of the judgment of exchangeability and even conditional exchangeability that should replace the misleading assertion of independence of various experts' opinions regarding uncertain situations. The application is technically rather complicated, and a basic understanding of exchangeability in its simple applications is presumed. One component of the analysis is novel. It suggests how we might
We introduce the New Zealand Whale Stranding Data Base, and use it to study the synchronicity of ... more We introduce the New Zealand Whale Stranding Data Base, and use it to study the synchronicity of Pilot whale Globicephala melaena stranding dates with the phases of the moon. We construct a data sequential conditioning procedure that distinguishes the extent to which the evidence supports three stranding theories: that stranding behaviour is uniform with respect to lunar phases; that stranding activity is procyclic with the full moon or the new moon; or that strandings are procyclic with the spring tides (as opposed to neap tides) which occur just after both the full moon and the new moon, when the difference between the high tide and the low tide is the greatest. Forecast distributions are based on judgments of exchangeability of lunar date sequences, and on mixing distributions that are themselves mixtures of one and two-cycle trigonometric distributions allowing phase shifts. Statistical results based on 49 sharply dated strandings since 1970 show the likelihood function is located near extreme amplitude values for both one and two-cycle distributions. The mixing proportionality factor is about one-half. A graphical method is used to exhibit the likelihood information regarding four parameters that are involved. The statistical analysis follows the operational subjective method developed in the writings of Bruno de Finetti. Subjectivist formulation of the statistical analysis allows a natural evaluation of the strength of the evidence in the data relative to theoretical evaluations of the scale of physical forces in the stranding phenomenon complex. Posterior predictive probabilities are computed based upon three different theory-based initial mixing functions.
We resolve a useful formulation of the question how a statistician can coherently incorporate the... more We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert's probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise.
The British Journal for the Philosophy of Science, 1984
Recent discussion of the calibration of probability assessments is related to the earlier influen... more Recent discussion of the calibration of probability assessments is related to the earlier influential attitudes of Frechet. The limiting frequency criterion of good calibration is criticised as being of no relevance to the evaluation of the probability of any event. An operational definition of good calibration is proposed which treats calibration properties as characteristics of the assessor's entire body of opinion, not of opinion about some particular event or events. In these terms a result is shown which says that every coherent opinion distribution is well-calibrated. Scoring rule representations as calibration plus refinement scores are discussed. It is proposed diat a subjectivist could well attribute particular bad experiences to 'luck', and must balance this sense of luck with a sense of the inadequacy of his/her own knowledge when evaluating previous knowledge and experience.
Re-sightings of photographically identified individuals were used to estimate survival rates for ... more Re-sightings of photographically identified individuals were used to estimate survival rates for a free-living population of Hector's dolphins Cephalorhynchus hectori, a species endemic to New Zealand waters. Most individuals were identified from injuries to the dorsal fin. Consequently, the photographic catalog contained very few young individuals. Our analysis included no newborn calves or yearlings, and provided estimates of survival rates only after the first year of life. We used two complementary methods for calculating survival rates: a modified Jolly-Seber model, and a simpler method which corrects in a more explicit way for individual dolphins being alive but not sighted. Selection of the most reliable subset of the data had a greater effect on computed survival rates than did the difference between the two methods. We conclude that careful inspection of resighting data before analysis, and, if necessary, selection of a subset, is very important in studies of this kind. Survival rate estimates came from a population which was subject to relatively heavy mortality from gillnet entanglement. Standard errors of the survival rate estimates have been used to assess the conditional probability of population decline given three fertility scenarios. The high probability that the Banks Peninsula Hector's dolphin population was decreasing during the study period (0.78 to 0.99) suggests that gillnet entanglement constituted a serious risk to this population.
This paper intends to show how the fabled violation of Bell's inequality by the probabilistic spe... more This paper intends to show how the fabled violation of Bell's inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein's principle of local realism, the inequality pertains to a li
I provide a critical reassessment of David Mermin's influential and misleading parable, "Quantum ... more I provide a critical reassessment of David Mermin's influential and misleading parable, "Quantum Mysteries for Anyone", identifying its errors and resolving them with a complete analysis of the quantum experiment it is meant to portray. Accessible to popular readership and requiring no knowledge of quantum physics at all, his exposition describes the curious behaviour of a machine that is designed to parody the empirical results of quantum experiments monitoring the spins of a pair of electrons under various conditions. The mysteries are said to unfold from contradictory results produced by a signal process that is proposed to explain them. I find that these results derive from a mathematical error of neglect, coupled with a confusion of two distinct types of experiments under consideration. One of these, a gedankenexperiment, provides the context in which the fabled defiance of Bell's inequality is thought to emerge. The errors are corrected by the recognition of functional relations embedded within the experimental conditions that have been long unnoticed. A Monte Carlo simulation of results in accord with the actual abstemious claims of quantum theory supports probability values that Mermin decries as unwarranted. However, the distribution it suggests is not definitive, in accord with the expressed agnostic position of quantum theory regarding measurements that cannot be executed. Bounding quantum probabilities are computed for the results of the gedankenexperiment relevant to Bell's inequality which inspired the parable. The problem is embedded in a 3 × 3 design of Stern-Gerlach magnet orientations at two observation stations. Computational resolution on the basis of Bruno de Finetti's fundamental theorem of probability requires the evaluation of a battery of three paired linear programming problems. Though technicalities are ornate, the message is clear. There are no mysteries of quantum mechanics that derive from mistaken understandings of Bell's inequality… for anyone.
Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, 1995
ABSTRACT Our focus is on the structure of conditional quantities and conditional prevision assert... more ABSTRACT Our focus is on the structure of conditional quantities and conditional prevision assertions which are crucial to implementing the fundamental theorem of prevision as an “inference engine”, foreseen in principle and constructed in simple forms for more than a century. I argue that conditional quantities do entail a minimal logical structure as required by the principle of coherency. But attempts to identify a more extensive structure of a logic of conditional events via many-valued logical functions are misdirected. The equivalent roles of arithmetic and of many-valued logics in generating a function space of object quantities are made apparent.
This article displays an application of the statistical method moti- vated by Bruno de Finetti... more This article displays an application of the statistical method moti- vated by Bruno de Finetti's operational subjective theory of probability. We use exchangeable forecasting distributions based on mixtures of linear com- binations of exponential power (EP) distributions to forecast the sequence of daily rates of return from the Dow-Jones index of stock prices over a 20 year period. The operational subjective statistical method for comparing distributions is quite different from that commonly used in data analysis, because it rejects the basic tenets underlying the practice of hypothesis test- ing. In its place, proper scoring rules for forecast distributions are used to assess the values of various forecasting strategies. Using a logarithmic scoring rule, we find that a mixture linear combination of EP distributions scores markedly better than does a simple mixture over the EP family, which scores much better than does a simple Normal mixture. Surprisingly, a mix- ture over a l...
A predictive distribution over a sequence of N + 1 events is said to be "frequency mimicking" whe... more A predictive distribution over a sequence of N + 1 events is said to be "frequency mimicking" whenever the probability for the final event conditioned on the outcome of the first N events equals the relative frequency of successes among them. Infinitely extendible exchangeable distributions that universally inhere this property are known to have several annoying concomitant properties. We motivate frequency mimicking assertions over a limited subdomain in practical problems of finite inference, and we identify their computable coherent implications. We provide some computed examples using reference distributions, and we introduce computational software to generate any specification. The software derives from an inversion of the finite form of the exchangeability representation theorem. Three new theorems delineate the extent of the usefulness of such distributions, and we show why it may not be appropriate to extend the frequency mimicking assertions for a specified value of N to any arbitrary larger size of N. The constructive results identify the source and structure of "adherent masses" in the limit of a sequence of finitely additive distributions. Appendices develop a novel geometrical representation of conditional probabilities which illuminate the analysis.
I reassess the gedankenexperiment of Greenberger, Horne, Shimony, and Zeilinger after twenty-five... more I reassess the gedankenexperiment of Greenberger, Horne, Shimony, and Zeilinger after twenty-five years, finding their influential claim to the discovery of an inconsistency inherent in high dimensional formulations of local realism to arise from a fundamental error of logic. They manage this by presuming contradictory premises: that a specific linear combination of four angles involved in their proposed parallel experiments on two pairs of electrons equals both π and 0 at the same time. Ignoring this while presuming the contradictory implications of these two conditions, they introduce the contradiction themselves. The notation they use in their “derivation” is not sufficiently ornate to represent the entanglement in the double electron spin pair problem they design, confounding their error. The situation they propose actually motivates only an understanding of the full array of symmetries involved in their problem. In tandem with the error now recognised in the supposed defiance o...
The refinement axiom for entropy has been provocative in providing foundations of information the... more The refinement axiom for entropy has been provocative in providing foundations of information theory, recognised as thoughtworthy in the writings of both Shannon and Jaynes. A resolution to their concerns has been provided recently by the discovery that the entropy measure of a probability distribution has a dual measure, a complementary companion designated as “extropy”. We report here the main results that identify this fact, specifying the dual equations and exhibiting some of their structure. The duality extends beyond a simple assessment of entropy, to the formulation of relative entropy and the Kullback symmetric distance between two forecasting distributions. This is defined by the sum of a pair of directed divergences. Examining the defining equation, we notice that this symmetric measure can be generated by two other explicable pairs of functions as well, neither of which is a Bregman divergence. The Kullback information complex is constituted by the symmetric measure of en...
ABSTRACT We study de Finetti’s “fundamental theorem of probability”, reformulated in the finite c... more ABSTRACT We study de Finetti’s “fundamental theorem of probability”, reformulated in the finite case as a computable linear programming problem. The theorem is substantially extended, and shown to have important implications for the subjective theory of statistical inference. It supports an operational meaning for the partial assertion of prevision via asserted bounds. We extend the theorem to apply to quantities more general than events, to allow bounds and orderings on previsions as the input to the programming problem, and even to allow bounds on conditional previsions as the input and output. In a philosophical discussion, prevision is explicitly recognized as a completion of the notion of logical assertion, introduced by Frege. Partial prevision assertions then allow the representation of weaker forms of uncertain knowledge, spanning a range of possible numerical values for a quantity between the assertion of a precise prevision value and the mere unassertive contemplation of the quantity.
All statistical measurements which represent the values of useful unknown quantities have a realm... more All statistical measurements which represent the values of useful unknown quantities have a realm that is both finite and discrete. Thus our uncertainties about any measurement can be represented by discrete probability mass functions. Nonetheless, common statistical practice treats probability distributions as representable by continuous densities or mixture densities. Many statistical problems involve the analysis of sequences of observations that the researcher regards exchangeably. Often we wish to find a joint probability mass function over X 1 , X 2 ,. .. , X n , with interim interest in the sequence of updated probability mass functions f (x i+1 | X i = x i) for i = 1, 2,. .. , n − 1. We investigate how well continuous conjugate theory can approximate real discrete mass functions in various measurement settings. Interest centres on approximating digital Normal mass functions and digital parametric mixtures with continuous Mixture Normal and Normal-Gamma Mixture Normal distributions for such items as E(X i+1 | X i = x i) and V (X i+1 | X i = x i). Digital mass functions are generated by specifying a finite realm of measurements for a quantity of interest, finding a density value of some specified function at each point, and then normalising the densities over the realm to generate mass values. Both a digitised prior mixing mass function and digitised information transfer function are generated and used, via Bayes' Theorem, to compute posterior mass functions. Approximating posterior densities using continuous conjugate theory are evaluated, and the two sets of results compared.
Mathematical models for handling partial knowledge in artificial intelligence, 1995
The operational subjective theory of probability and statistics recognizes Bruno de Finetti’s fun... more The operational subjective theory of probability and statistics recognizes Bruno de Finetti’s fundamental theorem of prevision as providing the basis for any coherent expert system of inferences from assertions of partial knowledge. One session of our workshop was devoted to understanding the formulation and computational structure of this theorem, as explained in the articles1 by Lad, Dickey, and Rahman (1990, 1992). In a word, the fundamental theorem of prevision identifies numerically the coherent implications of any partial prevision assertions (including interval probabilities) for any other cohering prevision or conditional prevision whatsoever. The present paper was discussed at a second session, and presumes the reader’s familiarity with these expositions. For the purpose of self-containment here, a brief summary of the definitions, notation, and theorems that lead to the fundamental theorem of prevision in a finite, discrete context, and a concise statement of the theorem in both its parts are appended to the end of this article. They are excerpted from the forthcoming text by Lad (1994). Included in the excerpt are definitions and a basic theorem pertaining to the judgment to regard a sequence of quantities exchangeably, which are also relevant to developments in the present article.
Applied Bayesian Statistical Studies in Biology and Medicine, 2004
Attending the Asti summer school where the papers in this book were presented were some thirty gr... more Attending the Asti summer school where the papers in this book were presented were some thirty graduate students writing empirical theses using applied statistics. Their topics ranged from the monitoring of commercial fishing stock around Sardinia, to learning about the situation of current Nigerian immigrants to Italy, and to comparing the nutrition and weight gain of children of Chinese immigrants in Italy with children of their non-emigrating cohort in a Chinese village. The thesis students were attending the course to improve their practical statistical skills. Having been involved in the modern development of subjective Bayesian methods for some thirty years myself, I was somewhat surprised by how little was known of these major developments which have transformed statistical thinking throughout the world over this period. They had received major impetus from the research of the noted Italian probabilist, Bruno de Finetti, and his followers.
We consider how to calculate the probability that the sum of the product of variables assessed wi... more We consider how to calculate the probability that the sum of the product of variables assessed with a Normal distribution is negative. The analysis is motivated by a specific problem in electrical engineering. To resolve the problem, two distinct steps are required. First, we consider ways in which we can assess the distribution for the product of two Normally distributed variables. Three different methods are compared: a numerical methods approximation, which involves implementing a numerical integration procedure on MATLAB, a Monte Carlo construction and an approximation to the analytic result using the Normal distribution. The second step considers how to assess the distribution for the sum of the products of two Normally distributed variables by applying the Convolution Formula. Finally, the two steps are combined to compute the distribution for the sum of products of Normally distributed variables, thus to calculate the probability that this sum of products is negative. The pro...
We re-examine the procedure of median decision making in the context of radiological determi- nat... more We re-examine the procedure of median decision making in the context of radiological determi- nation of asbestosis by three B-readers. Our assessment addresses the specificity, sensitivity and predictive values from this procedure compared to merely an individual radiologist's diagnosis. Con- ditional exchangeability of the radiologists' classifications is recognised as more appropriate than independence which is often presumed. The framework of de Finetti's fundamental theorem of prob- ability makes the analysis tractable when it is formulated in terms of a linear programming problem, yielding coherent bounds on probabilities of interest even when a complete distribution is not spec- ified. Further natural assertions motivate a partial ordering of conditional probabilities. In this context the computation of bounds develops into a quadratic programming problem. Using sensible assertions, the median decision procedure is found to be relatively weaker than has been though...
We present the technical details of the logitnormal distribution of a variable supported on the o... more We present the technical details of the logitnormal distribution of a variable supported on the open unit-interval. Although this distribution has been used widely in a variety of sta-tistical applications under a transformation, its direct analysis has not been exposited. The density function can be expressed algebraically, and the moments can be computed easily by numerical integration as functions of the parameters of the transformed normal distribution. The family of densities has characteristics different from the Beta family especially near 0 and 1, although some members of each family resemble each other. Limiting members of the family provide examples of distributions with adherent masses at 0 and 1. We also examine the bivariate exchangeable logitnormal distribution.
This article provides a completion to theories of information based on entropy, resolving a longs... more This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback-Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the L 2 metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.
Annals of Mathematics and Artificial Intelligence, 2002
This paper displays the productive role of the judgment of exchangeability and even conditional e... more This paper displays the productive role of the judgment of exchangeability and even conditional exchangeability that should replace the misleading assertion of independence of various experts' opinions regarding uncertain situations. The application is technically rather complicated, and a basic understanding of exchangeability in its simple applications is presumed. One component of the analysis is novel. It suggests how we might
We introduce the New Zealand Whale Stranding Data Base, and use it to study the synchronicity of ... more We introduce the New Zealand Whale Stranding Data Base, and use it to study the synchronicity of Pilot whale Globicephala melaena stranding dates with the phases of the moon. We construct a data sequential conditioning procedure that distinguishes the extent to which the evidence supports three stranding theories: that stranding behaviour is uniform with respect to lunar phases; that stranding activity is procyclic with the full moon or the new moon; or that strandings are procyclic with the spring tides (as opposed to neap tides) which occur just after both the full moon and the new moon, when the difference between the high tide and the low tide is the greatest. Forecast distributions are based on judgments of exchangeability of lunar date sequences, and on mixing distributions that are themselves mixtures of one and two-cycle trigonometric distributions allowing phase shifts. Statistical results based on 49 sharply dated strandings since 1970 show the likelihood function is located near extreme amplitude values for both one and two-cycle distributions. The mixing proportionality factor is about one-half. A graphical method is used to exhibit the likelihood information regarding four parameters that are involved. The statistical analysis follows the operational subjective method developed in the writings of Bruno de Finetti. Subjectivist formulation of the statistical analysis allows a natural evaluation of the strength of the evidence in the data relative to theoretical evaluations of the scale of physical forces in the stranding phenomenon complex. Posterior predictive probabilities are computed based upon three different theory-based initial mixing functions.
We resolve a useful formulation of the question how a statistician can coherently incorporate the... more We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert's probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise.
The British Journal for the Philosophy of Science, 1984
Recent discussion of the calibration of probability assessments is related to the earlier influen... more Recent discussion of the calibration of probability assessments is related to the earlier influential attitudes of Frechet. The limiting frequency criterion of good calibration is criticised as being of no relevance to the evaluation of the probability of any event. An operational definition of good calibration is proposed which treats calibration properties as characteristics of the assessor's entire body of opinion, not of opinion about some particular event or events. In these terms a result is shown which says that every coherent opinion distribution is well-calibrated. Scoring rule representations as calibration plus refinement scores are discussed. It is proposed diat a subjectivist could well attribute particular bad experiences to 'luck', and must balance this sense of luck with a sense of the inadequacy of his/her own knowledge when evaluating previous knowledge and experience.
Re-sightings of photographically identified individuals were used to estimate survival rates for ... more Re-sightings of photographically identified individuals were used to estimate survival rates for a free-living population of Hector's dolphins Cephalorhynchus hectori, a species endemic to New Zealand waters. Most individuals were identified from injuries to the dorsal fin. Consequently, the photographic catalog contained very few young individuals. Our analysis included no newborn calves or yearlings, and provided estimates of survival rates only after the first year of life. We used two complementary methods for calculating survival rates: a modified Jolly-Seber model, and a simpler method which corrects in a more explicit way for individual dolphins being alive but not sighted. Selection of the most reliable subset of the data had a greater effect on computed survival rates than did the difference between the two methods. We conclude that careful inspection of resighting data before analysis, and, if necessary, selection of a subset, is very important in studies of this kind. Survival rate estimates came from a population which was subject to relatively heavy mortality from gillnet entanglement. Standard errors of the survival rate estimates have been used to assess the conditional probability of population decline given three fertility scenarios. The high probability that the Banks Peninsula Hector's dolphin population was decreasing during the study period (0.78 to 0.99) suggests that gillnet entanglement constituted a serious risk to this population.
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