Subjective Probability Distributions Research Articles

A cortex-like learning machine for temporal hierarchical pattern clustering, detection, and recognition

A learning machine, called a clustering interpreting probabilistic associative memory (CIPAM), is proposed. CIPAM consists of a clusterer and an interpreter. The clusterer is a recurrent hierarchical neural network of unsupervised processing units (UPUs). The interpreter is a number of supervised processing units (SPUs) that branch out from the clusterer. Each processing unit (PU), UPU or SPU, comprises “dendritic encoders” for encoding inputs to the PU, “synapses” for storing resultant codes, a “nonspiking neuron” for generating inhibitory graded signals to modulate neighboring spiking neurons, “spiking neurons” for computing the subjective probability distribution (SPD) or the membership function, in the sense of fuzzy logic, of the label of said inputs to the PU and generating spike trains with the SPD or membership function as the firing rates, and a masking matrix for maximizing generalization. While UPUs employ unsupervised covariance learning mechanisms, SPUs employ supervised ones. They both also have unsupervised accumulation learning mechanisms. The clusterer of CIPAM clusters temporal and spatial data. The interpreter interprets the resultant clusters, effecting detection and recognition of temporal and hierarchical causes.

Inferring beliefs as subjectively imprecise probabilities

We propose a method for estimating subjective beliefs, viewed as a subjective probability distribution. The key insight is to characterize beliefs as a parameter to be estimated from observed choices in a well-defined experimental task and to estimate that parameter as a random coefficient. The experimental task consists of a series of standard lottery choices in which the subject is assumed to use conventional risk attitudes to select one lottery or the other and then a series of betting choices in which the subject is presented with a range of bookies offering odds on the outcome of some event that the subject has a belief over. Knowledge of the risk attitudes of subjects conditions the inferences about subjective beliefs. Maximum simulated likelihood methods are used to estimate a structural model in which subjects employ subjective beliefs to make bets. We present evidence that some subjective probabilities are indeed best characterized as probability distributions with non-zero variance.

A Low-Order Model of Biological Neural Networks

A biologically plausible low-order model (LOM) of biological neural networks is proposed. LOM is a recurrent hierarchical network of models of dendritic nodes and trees; spiking and nonspiking neurons; unsupervised, supervised covariance and accumulative learning mechanisms; feedback connections; and a scheme for maximal generalization. These component models are motivated and necessitated by making LOM learn and retrieve easily without differentiation, optimization, or iteration, and cluster, detect, and recognize multiple and hierarchical corrupted, distorted, and occluded temporal and spatial patterns. Four models of dendritic nodes are given that are all described as a hyperbolic polynomial that acts like an exclusive-OR logic gate when the model dendritic nodes input two binary digits. A model dendritic encoder that is a network of model dendritic nodes encodes its inputs such that the resultant codes have an orthogonality property. Such codes are stored in synapses by unsupervised covariance learning, supervised covariance learning, or unsupervised accumulative learning, depending on the type of postsynaptic neuron. A masking matrix for a dendritic tree, whose upper part comprises model dendritic encoders, enables maximal generalization on corrupted, distorted, and occluded data. It is a mathematical organization and idealization of dendritic trees with overlapped and nested input vectors. A model nonspiking neuron transmits inhibitory graded signals to modulate its neighboring model spiking neurons. Model spiking neurons evaluate the subjective probability distribution (SPD) of the labels of the inputs to model dendritic encoders and generate spike trains with such SPDs as firing rates. Feedback connections from the same or higher layers with different numbers of unit-delay devices reflect different signal traveling times, enabling LOM to fully utilize temporally and spatially associated information. Biological plausibility of the component models is discussed. Numerical examples are given to demonstrate how LOM operates in retrieving, generalizing, and unsupervised and supervised learning.

Eliciting probabilistic expectations with visual aids in developing countries: how sensitive are answers to variations in elicitation design?

AbstractEliciting subjective probability distributions in developing countries is often based on visual aids such as beans to represent probabilities and intervals on a sheet of paper to represent the support. We conduct an experiment in India which tests the sensitivity of elicited expectations to variations in three facets of the elicitation methodology: the number of beans, the design of the support (predetermined or self‐anchored), and the ordering of questions. Our results show remarkable robustness to variations in elicitation design. Nevertheless, the added precision offered by using more beans and a larger number of intervals with a predetermined support improves accuracy. Copyright © 2011 John Wiley & Sons, Ltd.

An adjustment scheme for nonlinear pricing problem with two buyers

We examine a contracting problem with asymmetric information in a monopoly pricing setting. Traditionally, the problem is modeled as a one-period Bayesian game, where the incomplete information about the buyers’ preferences is handled with some subjective probability distribution. Here we suggest an iterative online method to solve the problem. We show that, when the buyers behave myopically, the seller can learn the optimal tariff by selling the product repeatedly. In a practical modification of the method, the seller offers linear tariffs and adjusts them until optimality is reached. The adjustment can be seen as gradient adjustment, and it can be done with limited information and so that it benefits both the seller and the buyers. Our method uses special features of the problem and it is easily implementable.

Comparing the Point Predictions and Subjective Probability Distributions of Professional Forecasters

We use data from the Survey of Professional Forecasters (SPF) to compare point predictions of gross domestic product (GDP) growth and inflation with the subjective probability distributions held by forecasters. We find that most SPF point predictions are quite close to the central tendencies of forecasters subjective distributions tend to be asymmetric, with SPF forecasters tending to report point predictions that give a more favorable view of the economy than do their subjective means/medians/modes.

Measuring Consumer Uncertainty about Future Inflation

Survey measures of consumer inflation expectations have an important shortcoming in that, while providing useful summary measures of the distribution of point forecasts across individuals, they contain no direct information about an individual’s uncertainty about future inflation. The latter is important not only for forecasting inflation and other macroeconomic outcomes, but also for assessing a central bank’s credibility and effectiveness of communication. This paper explores the feasibility of eliciting individual consumers’ subjective probability distributions of future inflation outcomes.In November 2007, we began administering web-based surveys to participants in RAND’s American Life Panel. In addition to their point predictions, respondents were asked for their subjective assessments of the percentage chance that inflation will fall in each of several predetermined intervals. We find that our measures of individual forecast densities and uncertainty are internally consistent and reliable. Those who are more uncertain about year-ahead price inflation are also generally more uncertain about longer term price inflation and future wage changes. We find also that participants expressing higher uncertainty in their density forecasts make larger revisions to their point forecasts over time. Measures of central tendency derived from individual density forecasts are highly correlated with point forecasts, but they usually differ, often substantially, at the individual level.Finally, we relate our direct measure of aggregate consumer uncertainty to a more conventional approach that uses disagreement among individual forecasters, as seen in the dispersion of their point forecasts, as a proxy for forecast uncertainty. Although the two measures are positively correlated, our results suggest that disagreement and uncertainty are distinct concepts, both relevant to the analysis of inflation expectations.

Eliciting Subjective Probabilities in Internet Surveys.

Individuals' subjective expectations are important in explaining heterogeneity in individual choices, but their elicitation poses some challenges, in particular when one is interested in the subjective probability distribution of an individual. We have developed an innovative visual representation for Internet surveys that has some advantages over previously used formats. In this paper we present our findings from testing this visual representation in the context of individuals' Social Security expectations. Respondents are asked to allocate a total of 20 balls across seven bins to express what they believe the chances to be that their future Social Security benefits would fall into any one of those bins. Our data come from the Internet Survey of respondents to the Health and Retirement Study, a representative survey of the U.S. population age 51 and older. To contrast the results from the visual format with a previously used format we divided the sample into two random groups and administered both, the visual format and the more standard percent chance format. Our findings suggest that the main advantage of the visual format is that it generates usable answers for virtually all respondents in the sample while in the percent chance format a significant fraction (about 20 percent) of responses is lost due to inconsistencies. Across various other dimensions the visual format performs similarly to the percent chance format, leading us to conclude that the bins-and-balls format is a viable alternative that leads to more complete data.

Scoring Rules, Generalized Entropy, and Utility Maximization

Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative entropy), decision analysis (where new information can lead to improved decisions), and finance (where investors optimize portfolios based on their private information and risk preferences). In this paper, we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well-known generalized entropies and utility functions, shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization. In particular, we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy (divergence) with convenient properties, and they also correspond exactly to the solutions of expected utility maximization problems in which a risk-averse decision maker whose utility function belongs to the linear-risk-tolerance family interacts with a risk-neutral betting opponent or a complete market for contingent claims in either a one-period or a two-period setting. When the market is incomplete, the corresponding problems of maximizing linear-risk-tolerance utility with the risk-tolerance coefficient β are the duals of the problems of minimizing the pseudospherical or power divergence of order β between the decision maker's subjective probability distribution and the set of risk-neutral distributions that support asset prices.

Improving the way we think about projecting future energy use and emissions of carbon dioxide

A variety of decision makers need projections of future energy demand, CO2 emissions and similar factors that extend many decades into the future. The past performance of such projections has been systematically overconfident. Analysts have often used scenarios based on detailed story lines that spell out "plausible alternative futures" as a central tool for evaluating uncertainty. No probabilities are typically assigned to such scenarios. We argue that this practice is often ineffective. Rather than expanding people's judgment about the range of uncertainty about the future, scenario-based analysis is more likely to lead to systematic overconfidence, to an underestimate of the range of possible future outcomes. We review relevant findings from the literature on human judgment under uncertainty and discuss their relevance to the task of making probabilistic projections. The more detail that one adds to the story line of a scenario, the more probable it will appear to most people, and the greater the difficulty they likely will have in imagining other, equally or more likely, ways in which the same outcome could be reached. We suggest that scenario based approaches make analysts particularly prone to such cognitive biases, and then outline a strategy by which improved projections, tailored to the needs of specific decision makers, might be developed. For those of us who work on climate and energy policy it would be extremely useful to be able to predict a few simple things such as the future demand for energy and the future mix of energy technologies over the coming decades—if not as sharp point estimates, then at least as well-calibrated subjective probability distributions. However, the track-record of past efforts to make such predictions is anything but

Entrepreneuring as a puzzle: an attempt to its explanation with truncation of subjective probability distribution of prospects

AbstractThis research presents entrepreneuring as a puzzle where entrepreneurs venture at a risk‐return level that is worse than that of the private equity index and much worse than the public equity index. Under subjective rationality, we explain this puzzle by the possibility that entrepreneurs configure their outlook by dismissing some possible future performance states. Their forward view reflects a truncated and conditional distribution with overcredence. The unfavorable odds under a nontruncated distribution become very attractive after truncation. Copyright © 2008 Strategic Management Society.

HYPOTHESIS TESTING, STATISTICAL POWER, AND CONFIDENCE LIMITS IN THE PRESENCE OF EPISTEMIC UNCERTAINTY

Hypothesis testing, statistical power, and confidence limits are concepts from classical statistics that require data from observations. In some important recent applications some of the data are not observational but are reconstructed by computer models. There is generally epistemic uncertainty in model formulations, as well as in parameter and input values. The resulting epistemic uncertainty of the reconstructed data is determined by an uncertainty analysis and is expressed by subjective probability distributions. Sometimes only the mean or median values of the distributions are used in the concepts mentioned above, which hides the uncertainty of the data thereby rendering misleading results. Misleading results are also obtained if the epistemic uncertainty of the data is combined incorrectly with the stochastic variability of the outcome of the actual random complex concerned. This paper argues that an uncertainty analysis of the application of classical statistical concepts is essentially the correct way of dealing with the epistemic uncertainty of the data. A practical example serves as an illustration.

Subjective Beliefs and Schooling Decisions

This paper considers the estimation of sequential schooling decisions made by agents who are endowed with subjective beliefs about their own ability. I use unique Italian panel data which provide information on i) the curvature of the per-period utility function, ii) schooling decisions, iii) post-schooling earnings, in order to estimate the future component of the differences in intertemporal utilities of school and work independently from the component, (as in Geweke and Keane, 1995, 2001), and evaluate the importance of present bias. Under certain conditions, which include imposing equality between the modal belief and true ability, I recover individual specific subjective probability distributions. I estimate both the degree of confidence (a measure of spread) and the incidence of over (and under) estimation. I find that the future component of intertemporal utilities dominates schooling decisions. I find a strong incidence of under-estimation among the more able and a much smaller incidence of over-estimation among the low ability group. At the medium ability spectrum, there is evidence of some over-estimation. The degree of confidence is high and implies that agents have a substantial amount of inside information (36% of the population act on a degenerate subjective distribution). Overall, the variance of the objective ability heterogeneity distribution is 4 times as large the variance of the distribution characterizing subjective beliefs.

Decisions Under Uncertainty (Decisiones Bajo Incertidumbre)

This is a course material from the book Managerial Decision Making Under Risk and Uncertainty. The book is originally in Spanish and is untitled as Decisiones empresariales bajo riesgo e incertidumbre. The level of the book is basic. We use very few mathematics and it is expected to be used by managers. In this third chapter we deal with the idea of uncertainty (not having a probability distribution assigned to the events). We use the basic ideas of game theory to introduce the reader into the strategies when dealing with uncertainty. We illustrate the different approaches in game theory: minimax, maxmax, minimum regret, Laplace method, etc. We also present a simple method to estimate the probability of an outcome and one method to estimate a subjective probability distribution.

HYPOTHESIS TESTING, STATISTICAL POWER, AND CONFIDENCE LIMITS IN THE PRESENCE OF EPISTEMIC UNCERTAINTY

Hypothesis testing, statistical power, and confidence limits are concepts from classical statistics that require data from observations. In some important recent applications some of the data are not observational but are reconstructed by computer models. There is generally epistemic uncertainty in model formulations, as well as in parameter and input values. The resulting epistemic uncertainty of the reconstructed data is determined by an uncertainty analysis and is expressed by subjective probability distributions. Sometimes only the mean or median values of the distributions are used in the concepts mentioned above, which hides the uncertainty of the data thereby rendering misleading results. Misleading results are also obtained if the epistemic uncertainty of the data is combined incorrectly with the stochastic variability of the outcome of the actual random complex concerned. This paper argues that an uncertainty analysis of the application of classical statistical concepts is essentially the correct way of dealing with the epistemic uncertainty of the data. A practical example serves as an illustration.

Addressing the uncertainties related to passive system reliability

Uncertainties involved in passive system operation play a crucial role in the relative reliability assessment endeavor. These uncertainties concern the deviations of the underlying physical principles from the expected behavior and are related to the onset of physical phenomena infringing the system performance or to the changes in the initial/boundary conditions. This issue is addressed by the development of subjective probability distributions relative to the physical modes of failure for a typical passive system for reactor decay heat removal, operating in natural circulation. Because of the limited availability of data, both experimental and operational, expert judgment and engineering assessment are used to characterize the present status of uncertainty with regard to a set of critical parameters driving the failure modes. This allows deriving a set of natural circulation failure probability distributions to quantitatively get information on system reliability.

Elicitation of Expert Judgments of Aerosol Forcing

A group of twenty-four leading atmospheric and climate scientists provided subjective probability distributions that represent their current judgment about the value of planetary average direct and indirect radiative forcing from anthropogenic aerosols at the top of the atmosphere. Separate estimates were obtained for the direct aerosol effect, the semi-direct aerosol effect, cloud brightness (first aerosol indirect effect), and cloud lifetime/distribution (second aerosol indirect effect). Estimates were also obtained for total planetary average forcing at the top of the atmosphere and for surface forcing. Consensus was strongest among the experts in their assessments of the direct aerosol effect and the cloud brightness indirect effect. Forcing from the semi-direct effect was thought to be small (absolute values of all but one of the experts' best estimates were ≤0.5 W/m2). There was not agreement about the sign of the best estimate of the semi-direct effect, and the uncertainty ranges some experts gave for this effect did not overlap those given by others. All best estimates of total aerosol forcing were negative, with values ranging between −0.25 W/m2 and −2.1 W/m2. The range of uncertainty that a number of experts associated with their estimates, especially those for total aerosol forcing and for surface forcing, was often much larger than that suggested in 2001 by the IPCC Working Group 1 summary figure (IPCC, 2001).

A methodology for evaluating risk and efficacy of weed management technologies

A stochastic simulation modeling framework was developed for measuring the impact of weed management technologies in terms of their risk and efficacy. The framework explicitly accounted for the variability of environmental conditions, which underpins risk, and its effect upon the weed population dynamics and crop yields. It was applied to wild oat and wild radish in spring wheat as case studies. Technologies considered included a postemergence herbicide, preseeding tillage, increased crop density, and a selective spray-topping (seed-head sterilizing) herbicide. All stages of the weed life cycle were specified as random variables on the basis of triangular probability distributions, which either were derived from daily environmental conditions or specified as a subjective probability distribution. By using probability density functions the study identified the risks to changes in the weed seed bank and weed density associated with various integrated weed management strategies. This approach not only quantified the probabilities associated with the different outcomes, but also identified how the probability distributions of outcomes were changed as a result of different technology combinations used in weed management strategies. For instance, strategies involving a selective spray-topping herbicide to control seed rain not only resulted in lower seed banks, but the range in possible values was considerably reduced, implying lower likelihood of a population increase. The use of such a risk framework not only benefits weed scientists in terms of designing more effective weed management technologies, but can also assist farmer adoption by being able to quantify the probabilities of success and failure of a technology.

RADIATION EFFECTIVENESS FACTORS FOR USE IN CALCULATING PROBABILITY OF CAUSATION OF RADIOGENIC CANCERS

This paper presents so-called radiation effectiveness factors that are intended to represent the biological effectiveness of different radiation types, relative to high-energy Co gamma rays, for the purpose of estimating cancer risks and probability of causation of radiogenic cancers in identified individuals. Radiation effectiveness factors are expressed as subjective probability distributions to represent uncertainty that arises from uncertainties in estimates of relative biological effectiveness obtained from radiobiological studies of stochastic endpoints, limited data on biological effectiveness obtained from human epidemiological studies, and other judgments involved in evaluating the applicability of available information to induction of cancers in humans. Primarily on the basis of reviews and evaluations of available data by experts, probability distributions of radiation effectiveness factors are developed for the following radiation types: neutrons of energy less than 10 keV, 10-100 keV, 0.1-2 MeV (including fission neutrons), 2-20 MeV, and greater than 20 MeV; alpha particles of any energy emitted by radionuclides; photons of energy 30-250 keV and less than 30 keV; and electrons of energy less than 15 keV. Photons of energy greater than 250 keV and electrons of energy greater than 15 keV are assumed to have the same biological effectiveness as reference Co gamma rays and are assigned a radiation effectiveness factor of unity, without uncertainty. For neutrons and alpha particles, separate probability distributions of radiation effectiveness factors are developed for solid tumors and leukemias, and small corrections to represent an inverse dose-rate effect are applied to those distributions in cases of chronic exposure. A radiation effectiveness factor different from unity for 15-60 keV electrons is discussed but is not adopted due to a lack of relevant radiobiological data. Radiation effectiveness factors presented in this paper are incorporated in the Interactive RadioEpidemiological Program and were developed for use by The National Institute for Occupational Safety and Health and U.S. Department of Labor in evaluating claims for compensation for radiogenic cancers by workers at U.S. Department of Energy facilities.

A Structured Expert Judgment Study for a Model of Campylobacter Transmission During Broiler‐Chicken Processing

A structured expert judgment study was organized to obtain input data for a microbial risk-assessment model describing the transmission of campylobacter during broiler-chicken processing in the Netherlands. More specially, the expert study was aimed at quantifying the uncertainty on input parameters of this model and focused on the contamination of broiler-chicken carcasses with campylobacter during processing. Following the protocol for structured expert judgment studies, expert assessments were elicited individually through subjective probability distribution functions. The classical model was used to aggregate the individual experts' distributions in order to obtain a single combined distribution per variable. Three different weighting schemes were applied, including equal weighting and performance-based weighting with and without optimalization of the combined distributions. The individual experts' weights were based on their performance on the seed variables. Results of the various weighting schemes are presented in terms of performance, robustness, and combined distributions of the seed variables and some of the query variables. All three weighting schemes had adequate performance, with the optimized combined distributions significantly outperforming both the equal weight and the nonoptimized combined distributions. Hence, this weighting scheme, having adequate robustness, was chosen for further processing of the results.