Frank Lad

Two Moments of the Logitnormal Distribution

We display the first two moment functions of the Logitnormal(μ, σ2) family of distributions, conveniently described in terms of the Normal mean, μ, and the Normal signal-to-noise ratio, μ/σ, parameters that generate the family. Long neglected on account of the numerical integrations required to compute them, awareness of these moment functions should aid the sensible interpretation of logistic regression statistics and the specification of “diffuse” prior distributions in hierarchical models, which can be deceiving. We also use numerical integration to compare the correlation between bivariate Logitnormal variables with the correlation between the bivariate Normal variables from which they are transformed.

Extropy: Complementary Dual of Entropy

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 L2 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.

Completing the logarithmic scoring rule for assessing probability distributions

We propose and motivate an expanded version of the logarithmic score for forecasting distributions, termed the Total Log score. It incorporates the usual logarithmic score, which is recognised as incomplete and has been mistakenly associated with the likelihood principle. The expectation of the Total Log score equals the Negentropy plus the Negextropy of the distribution. We examine both discrete and continuous forms of the scoring rule, and we discuss issues of scaling for scoring assessments. The analysis suggests the dual tracking of the quadratic score along with the usual log score when assessing the qualities of probability distributions. An application to the sequential scoring of forecast distributions for the daily rate of stock returns displays the usefulness of the proposal.

Numerical application of the fundamental theorem of prevision

The fundamental theorem of the operational subjective theory of probability, from the viewpoint established by Bruno de Finetti, provides the solutions to probability problems in terms of the results of a pair of specific linear or nonlinear programming problems. We state a general version of this theorem in a computationally feasible form and present numerical examples of its use. The examples display interesting extensions of the Bienayme-Chebyshev inequality and a variation on the Kolmogorov inequality in the context of finite discrete quantities. The Bienayme-Chebyshev application is extended to exemplify the use of a nonlinear programming algorithm to resolve a common question regarding coherent inference. In concluding discussion, we comment on the sizes of realistic problems and suggest a variety of applications for such computations, among them the safety assessment of complex engi-neering systems, the analysis of agricultural production statistics, and a synthesis of subjective judgments in macro...

Sequentially forecasting economic indices using mixture linear combinations of EP distributions.

This article displays an application of the statistical method moti- vated by Bruno de Finettis 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 linear combination of three Normal distributions also makes a substantial improvement over a simple Normal mixture, although it does not quite match the performance of even the simple EP mixture. All sub- stantive forecasting improvements become most marked after extreme tail phenomena were actually observed in the sequence, in particular after the abrupt drop in market prices in October, 1987. However, the improvements continue to be apparent over the long haul of 1985-2006 which has seen a number of extreme price changes. This result is supported by an analysis of the Negentropies embedded in the forecasting distributions, and a proper scoring analysis of these Negentropies as well.

An Application of Operational-Subjective Statistical Methods to Rational Expectations

This article presents a sequential scoring analysis of six econometric forecast distributions for the main components of the annual U.S. gross national product (GNP) accounts—nominal GNP, real GNP, and the implicit price deflator. Analysis of sequential forecasts is presented in terms of proper scoring rules. Computations relevant to the calibration and refinement properties of the forecast distributions are discussed. Annual data are studied for the period 1952–1982. The six forecast distributions are distinguished by the different stances they entail with respect to a subjectivist characterization of the rational-expectations hypothesis.

Coherency conditions for finite exchangeable inference

We idenify the invertible coherent functional relation between an array of asserted conditional probabilities and the probability distribution for the sum of events that are regarded exchangeably, in the regular case thatP(N N+1 |S N =a) ∈ (0, 1) for everya=0, 1, ...,N. The result is used to construct a useful algebraic and geometrical representation of all coherent inferences in the regular case, including those that are nonlinear in the sum of the conditioning events. The special case in which conditional probabilities mimic observed frequencies within (0, 1) receives an exact solution, which allows an easy interpretation of its surprising consequences. Finally, we introduce a new direction in research on prior opinion assessment that this approach, inverse to the usual one, suggests.

Coherent prevision as a linear functional without an underlying measure space: the purely arithmetic structure of logical relations among conditional quantities

My first two presentations at this workshop were oriented toward the theoretical construction and a substantive application of the operational subjective statistical outlook to numerically specified systems of uncertain knowledge. In this concluding discussion I shall emphasize the formal aspects of these activities that appear critically different from the structures associated with recent work on conditional events that derives its outlook at least implicitly from a measure theoretic tradition in probability theory. Specifically, our focus will be 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 shall 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 shall be made apparent. I shall again presume the reader’s familiarity with the detailed definitions and conceptual exposition appearing in the articles by Lad, Dickey, and Rahman (1990, 1992) which were discussed in my first session at this workshop. A brief review of relevant notation, definitions, and theorems appears in an Appendix to the article of Lad and Coope in the present volume.

Synchronicity of Whale Strandings with Phases of the Moon

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.

The Duality of Entropy/Extropy, and Completion of the Kullback Information Complex

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 entropy/extropy along with one of each of these three function pairs. It is intimately related to the total logarithmic score of two distinct forecasting distributions for a quantity under consideration, this being a complete proper score. The information complex is isomorphic to the expectations that the two forecasting distributions assess for their achieved scores, each for its own score and for the score achieved by the other. Analysis of the scoring problem exposes a Pareto optimal exchange of the forecasters’ scores that both are willing to engage. Both would support its evaluation for assessing the relative quality of the information they provide regarding the observation of an unknown quantity of interest. We present our results without proofs, as these appear in source articles that are referenced. The focus here is on their content, unhindered. The mathematical syntax of probability we employ relies upon the operational subjective constructions of Bruno de Finetti.