Estimating Subjective Probabilities

Subjective probabilities play a role in many economic decisions. There is a large theoretical literature on the elicitation of subjective probabilities, and an equally large empirical literature. However, there is a gulf between the two. The theoretical literature proposes a range of procedures that can be used to recover subjective probabilities, but stresses the need to make strong auxiliary assumptions or "calibrating adjustments" to elicited reports in order to recover the latent probability. With some notable exceptions, the empirical literature seems intent on either making those strong assumptions or ignoring the need for calibration. We illustrate how the joint estimation of risk attitudes and subjective probabilities using structural maximum likelihood methods can provide the calibration adjustments that theory calls for. This allows the observer to make inferences about the latent subjective probability, calibrating for virtually any well-specified model of choice under uncertainty. We demonstrate our procedures with experiments in which we elicit subjective probabilities. We calibrate the estimates of subjective beliefs assuming that choices are made consistently with expected utility theory or rank-dependent utility theory. Inferred subjective probabilities are significantly different when calibrated according to either theory, thus showing the importance of undertaking such exercises. Our findings also have implications for the interpretation of probabilities inferred from prediction markets.

Inferring Beliefs as Subjectively Uncertain 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.

Inferring Beliefs as Subjectively Uncertain 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

Estimating Subjective Probabilities

Subjective probabilities play a central role in many economic decisions, and act as an immediate confound of inferences about behavior, unless controlled for. Several procedures to recover subjective probabilities have been proposed, but in order to recover the correct latent probability one must either construct elicitation mechanisms that control for risk aversion, or construct elicitation mechanisms which undertake “calibrating adjustments” to elicited reports. We illustrate how the joint estimation of risk attitudes and subjective probabilities can provide the calibration adjustments that theory calls for. We illustrate this approach using data from a controlled experiment with real monetary consequences to the subjects. This allows the observer to make inferences about the latent subjective probability, under virtually any well-specified model of choice under subjective risk, while still employing relatively simple elicitation mechanisms

Estimating Aversion to Uncertainty

It is intuitive that decision-makers might have attitudes towards uncertainty just as they might have attitudes towards risk. However, it is only recently that this intuitive notion has been formalized and axiomatically characterized. We estimate the extent of uncertainty aversion in a manner that is parsimonious and consistent with theory. We demonstrate that one can jointly estimate attitudes towards uncertainty, attitudes towards risk, and subjective probabilities in a rigorous manner. Our structural econometric model constructively demonstrates the theoretical claims that it is possible to define uncertainty aversion in an empirically tractable manner. Our results show that attitudes towards risk and uncertainty can be different, qualitatively and quantitatively, and that allowing for these differences can have significant effects on inferences about subjective probabilities

Theory and Experiments

Subjective beliefs play a role in many economic decisions. There is a large theoretical literature on the elicitation of beliefs, and an equally large empirical literature. However, there is a gulf between the two. The theoretical literature proposes a range of procedures that can be used to recover beliefs, but stresses the need to make strong auxiliary assumptions or “calibrating adjustments” to elicited reports in order to recover the latent belief. With some notable exceptions, the empirical literature seems intent on either making those strong assumptions or ignoring the need for calibration. We make three contributions to bridge this gulf. First, we offer a general theoretical framework in which the belief elicitation task can be viewed as an exchange of state-dependent commodities between two traders. Second, we provide a specific elicitation procedure which has clear counterparts in field betting environments, and that is directly motivated by our theoretical framework. Finally, we illustrate how one can jointly estimate risk attitudes and subjective beliefs using structural maximum likelihood methods. This allows the observer to make inferences about the latent subjective belief, calibrating for virtually any well-specified model of choice under uncertainty. We demonstrate our procedures with an experiment in which we elicit subjective probabilities over three future events and one fact

Estimating Subjective Probabilities

Subjective probabilities play a role in many economic decisions. There is a large theoretical literature on the elicitation of subjective probabilities, and an equally large empirical literature. However, there is a gulf between the two. The theoretical literature proposes a range of procedures that can be used to recover subjective probabilities, but stresses the need to make strong auxiliary assumptions or “calibrating adjustments” to elicited reports in order to recover the latent probability. With some notable exceptions, the empirical literature seems intent on either making those strong assumptions or ignoring the need for calibration. We illustrate how one can jointly estimate risk attitudes and subjective probabilities using structural maximum likelihood methods. This allows the observer to make inferences about the latent subjective probability, calibrating for virtually any well-specified model of choice under uncertainty. We demonstrate our procedures with experiments in which we elicit subjective probabilities. We calibrate the estimates of subjective beliefs assuming that choices are made consistently with expected utility theory or rank-dependent utility theory. Inferred subjective probabilities are significantly different when calibrated according to either theory

Glacier-bed geomorphic processes and hydrologic conditions relevant to nuclear waste disposal

Characterizing glaciotectonic deformation, glacial erosion and sedimentation, and basal hydrologic conditions of ice sheets is vital for selecting sites for nuclear waste repositories at high latitudes. Glaciotectonic deformation is enhanced by excess pore pressures that commonly persist near ice sheet margins. Depths of such deformation can extend locally to a few tens of meters, with depths up to approximately 300 m in exceptional cases. Rates of glacial erosion are highly variable (0.05–15 mm a−1), but ratesa−1 are expected in tectonically quiescent regions. Total erosion probably not exceeding several tens of meters is expected during a glacial cycle, although locally erosion could be greater. Consolidation of glacial sediments that is less than expected from independent estimates of glacier thickness indicates that heads at the bases of past ice sheets were usually within 30% of the floatation value. This conclusion is reinforced by direct measurements of water pressure beneath portions of the West Antarctic ice sheet, which indicate average headsbed, despite thick ice at subfreezing temperatures. Therefore, in models of subglacial groundwater flow used to assess sites for nuclear waste repositories, a flux upper boundary condition based on water input from only basal melting will be far more uncertain than applying a hydraulic head at the upper boundary set equal to a large fraction of the floatation value

A review of the impacts of degradation threats on soil properties in the UK

National governments are becoming increasingly aware of the importance of their soil resources and are shaping strategies accordingly. Implicit in any such strategy is that degradation threats and their potential effect on important soil properties and functions are defined and understood. In this paper, we aimed to review the principal degradation threats on important soil properties in the UK, seeking quantitative data where possible. Soil erosion results in the removal of important topsoil and, with it, nutrients, C and porosity. A decline in soil organic matter principally affects soil biological and microbiological properties, but also impacts on soil physical properties because of the link with soil structure. Soil contamination affects soil chemical properties, affecting nutrient availability and degrading microbial properties, whilst soil compaction degrades the soil pore network. Soil sealing removes the link between the soil and most of the ‘spheres’, significantly affecting hydrological and microbial functions, and soils on re-developed brownfield sites are typically degraded in most soil properties. Having synthesized the literature on the impact on soil properties, we discuss potential subsequent impacts on the important soil functions, including food and fibre production, storage of water and C, support for biodiversity, and protection of cultural and archaeological heritage. Looking forward, we suggest a twin approach of field-based monitoring supported by controlled laboratory experimentation to improve our mechanistic understanding of soils. This would enable us to better predict future impacts of degradation processes, including climate change, on soil properties and functions so that we may manage soil resources sustainably

Snowball Earth climate dynamics and Cryogenian geology–geobiology

Geological evidence indicates that grounded ice sheets reached sea level at all latitudes during the long‐lived Sturtian (717–659 Ma) and Marinoan (ca 645–635 Ma) glaciations. Combined U-­‐Pb and Re-­‐Os geochronology suggests that the Sturtian glacial onset and both terminations were globally synchronous. Geochemical data imply that atmospheric pCO2 was 102x modern at the Marinoan termination, consistent with Snowball Earth hysteresis. Sturtian glaciation followed the breakup of a tropical supercontinent, and its onset coincided with the equatorial emplacement of a large igneous province. Modeling shows that the small thermal inertia of a globally frozen surface reverses the annual-mean Hadley circulation, resulting in equatorial net sublimation and net deposition elsewhere. Oceanic ice thickens, forming a sea glacier that flows gravitationally toward the equator, sustained by the hydrologic cycle and by basal freeze-on and melting. Tropical ice sheets flow faster as CO2 rises, but lose mass and become sensitive to orbital forcing. Dust accumulation in the equatorial zone engenders supraglacial oligotrophic meltwater ecosystems, favorable for cyanobacteria and many eukaryotes. Meltwater flushing through moulins enables organic burial and submarine deposition of subaerially-­‐erupted volcanic ash. The subglacial ocean is turbulent and well­‐mixed, in response to geothermal heating and conductive heat loss through the ice cover, increasing with latitude. Cap carbonates, unique to Snowball Earth terminations, are products of intense weathering and ocean stratification. Whole-­ocean warming and ice-sheet forebulge collapse allow marine coastal inundations to progress long after ice-sheet disappearance. The evolutionary legacy of Snowball Earth is perceptible in fossils and living organisms