Incremental conditioning of lower and upper probabilities

Lower probability models for uncertainty and nondeterministic processes

A probabilistic thunderstorm forecast contest has been hosted by the Department of Atmospheric and Environmental Sciences at the University at Albany since 1979. The contest involves issuing forecasts of the 24-h probability of thunder at 10 locations in the contiguous United States during the summer. A previous study by Bosart and Landin in 1994 used consensus forecasts from this contest to assess probabilistic thunder forecast skill during the 1983–93 period. Given the research advances and improvements in observation and modeling systems that have occurred since 1993, a renewed analysis of thunderstorm forecast skill in the contest is warranted. This study revisits the conclusions of the previous study and provides an updated assessment of the state of human and model thunderstorm forecast skill for the period from 2009 to 2022. Through the lens of this contest, probabilistic thunderstorm forecast skill has increased substantially in the past two decades. The 14-yr average skill from 2009 to 2022 was 34.2% (standard deviation of 5.7%) relative to climatology, an increase from 21.8% during the 1983–93 period. This increase in forecast skill is found to largely be a result of improved forecast confidence with continued forecast reliability. In the 2009–22 period, contest participants tended to forecast at the extreme ends of the probability distribution more often than in 1983–93, evidence of greater forecast confidence due to improved nowcasting and short-term forecasting tools. Significance Statement The purpose of this study is to examine how the ability of forecasters to predict the probability of thunderstorms has improved over time. To examine thunderstorm forecast skill, we analyze entries in a thunderstorm forecast contest. Our results demonstrate that probabilistic thunderstorm forecast skill has improved greatly in the past few decades. This skill increase occurred as forecasters have become more confident in issuing forecasts of either very high or very low thunder probabilities, although participants continue to overforecast the probability of thunder at low probabilities and underforecast the probability of thunder at high probabilities. These results demonstrate that research advances and improvements in technology are resulting in more skillful forecasts of hazardous weather such as thunderstorms.

On Keynes's Realization That His Concept of Uncertainty in the General Theory, Defined as an Inverse Function of Weight from the 'A Treatise on Probability', Would Not Have Any Impact at All on Economists Thinking

The paper proposes a parallel survey of revision and updating operations available in the probability theory and in the possibility theory frameworks. In these two formalisms the current state of knowledge is generally represented by a [0,1]-valued function whose domain is an exhaustive set of mutually exclusive possible states of the world. However, in possibility theory, the unit-interval can be viewed as a purely ordinal scale. Two general kinds of operations can be defined on this assignment function: conditioning, and imaging (or “projection”). the difference between these two operations is analogous to the one made between belief revision à la Gärdenfors and updating à la Katsuno and Mendelzon in the logical framework. In the probabilistic framework these two operations are respectively Bayesian conditioning and Lewis' imaging. Counterparts to these operations are presented for the possibilistic framework including the case of conditioning upon uncertain observations, and justifications are given which parallel the ones existing for the probabilistic operations. More particularly, it is recalled that possibilistic conditioning satisfies all the postulates proposed by Alchourrón, Gärdenfors and Makinson for belief revision (stated in possibilistic terms), and it is proved that possibilistic imaging satisfies all the postulates proposed by Katsuno and Mendelzon. the situation where our current knowledge is stated in terms of weighted logical propositions is discussed in connection to possibility theory. Revision in other more complex numerical formalisms, namely belief and plausibility functions, and upper and lower probabilities is also surveyed. Recent results on the revision of conditional knowledge bases are also reviewed. the frameworks of belief functions, upper and lower probabilities and conditional bases are more sophisticated than the previous ones because they enable to distinguish between factual evidence and generic knowledge in a cognitive state. This, framework leads to two forms of belief revision respectively taking care of the revision of evidence and the revision of knowledge. © 1994 John Wiley & Sons, Inc.

The paper proposes a new disclosure limitation procedure based on simulation. The key feature of the proposal is to protect actual microdata by drawing artificial units from a probability model, that is estimated from the observed data. Such a model is designed to maintain selected characteristics of the empirical distribution, thus providing a partial representation of the latter. The characteristics we focus on are the expected values of a set of functionss these are constrained to be equal to their corresponding sample averagess the simulated data, then, reproduce on average the sample characteristics. If the set of constraints covers the parameters of interest of a user, information loss is controlled for, while, as the model does not preserve individual values, re-identification attempts are impaired-synthetic individuals correspond to actual respondents with very low probability. Disclosure is mainly discussed from the viewpoint of record re-identification. According to this definition, as the pledge for confidentiality only involves the actual respondents, release of synthetic units should in principle rule out the concern for confidentiality. The simulation model is built on the Italian sample from the Community Innovation Survey (CIS). The approach can be applied in more generality, and especially suits quantitative traits. The model has a semi-parametric component, based on the maximum entropy principle, and, here, a parametric component, based on regression. The maximum entropy principle is exploited to match data traitss moreover, entropy measures uncertainty of a distribution: its maximisation leads to a distribution which is consistent with the given information but is maximally noncommittal with regard to missing information. Application results reveal that the fixed characteristics are sustained, and other features such as marginal distributions are well represented. Model specification is clearly a major points related issues are selection of characteristics, goodness of fit and strength of dependence relations.

Introduction

The Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection of “bad” events which are mostly independent and have low probability. A seminal algorithm of Moser and Tardos (J. ACM, 2010, 57, 11) (which we call the MT algorithm) gives nearly‐automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (Combinatorica, 1985, 5, 241–245); this is more powerful than alternate LLL criteria and also leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility. Third, we provide a derandomized version of the MT‐distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non‐repetitive vertex coloring, independent transversals, strong coloring, and other problems.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Deterministic algorithms for the Lovász Local Lemma: simpler, more general, and more parallelDavid G. HarrisDavid G. Harrispp.1744 - 1779Chapter DOI:https://doi.org/10.1137/1.9781611977073.71PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The Lovász Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection ℬ of “bad” events which are mostly independent and have low probability. In its simplest “symmetric” form, it asserts that whenever a bad-event has probability p and affects at most d bad-events, and epd < 1, then a configuration avoiding all ℬ exists. A seminal algorithm of Moser & Tardos (2010) (which we call the MT algorithm) gives nearly-automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (1985); this is more powerful than alternate LLL criteria and also removes a number of nuisance parameters and leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility in the functional form of the bad-events. Third, we provide a derandomized version of the MT-distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non-repetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

This paper addresses the problem of why the conditions under which standard proofs of the Dutch Book argument proceed should ever be met. In particular, the condition that there should be odds at which you would be willing to bet indifferently for or against are hardly plausible in practice, and relaxing it and applying Dutch book considerations gives only the theory of upper and lower probabilities. It is argued that there are nevertheless admittedly rather idealised circumstances in which the classic form of the Dutch Book argument is valid.

The SIPTA (Society for Imprecise Probability,Theory and Application) view of Keynes’s contributions to imprecise probability and application form a one-to-one, onto mapping from the claims of Henry E. Kyburg to what is accepted as being what Keynes’s contribution was. Thus, if one is familiar with Kyburg’s assessment of Keynes’s contributions, then one also knows what SIPTA’s views of Keynes’s contributions are. There is an extremely severe problem here, however, because Kyburg never read beyond chapter 6 in Keynes’s A Treatise on Probability. Kyburg was, therefore, ignorant about the many technical and mathematical contributions made to the theory of imprecise probability by Keynes, such as upper and lower probabilities, non additive probabilities,sub additive probabilities, interval valued estimates, decision weights, etc, in Parts II, III, IV and V of the A Treatise on Probability. Non additive probability appears in Keynes’s applications of non additive probability in his analysis of uncertainty in his theory of the liquidity preference approach to the rate of interest in the General Theory, where uncertainty was defined as an inverse function of the evidential weight of the argument in footnote 1 on page 148 of chapter 12. Keynes greatly emphasized the relative importance of his 'non-numerical' (interval) probabilities and weight of argument in his correspondence with Townshend in 1937-38. Kyburg’s view of Keynes’s contribution was to argue, based on Keynes’s verbal discussions on pp.30-34 of chapter III of Keynes’s A Treatise on Probability, that Keynes had been the first to (a) define a partial order and (b) the first to argue that all probabilities were not comparable and measurable. Kyburg also acknowledged Keynes’s contribution in chapter 6 concerning the weight of the argument as it related to the Bayesian refusal to give any importance to questions dealing with the relative strength of the evidence of different probabilities, i.e. whether the evidentiary base supporting a probability estimate was composed of strong evidence or made up of weak evidence. We will conclude that the above assessment made by Kyburg is identical to the SIPTA assessment made from 1999 (Kyburg’s 1999 assessment for SIPTA) to the last SIPTA meeting in 2019.

This article presents a new theory of subjective probability according to which different descriptions of the same event can give rise to different judgments. The experimental evidence confirms the major predictions of the theory. First, judged probability increases by unpacking the focal hypothesis and decreases by unpacking the alternative hypothesis. Second, judged probabilities are complementary in the binary case and subadditive in the general case, contrary to both classical and revisionist models of belief. Third, subadditivity is more pronounced for probability judgments than for frequency judgments and is enhanced by compatible evidence. The theory provides a unified treatment of a wide range of empirical findings. It is extended to ordinal judgments and to the assessment of upper and lower probabilities. Both laypeople and experts are often called upon to evaluate the probability of uncertain events such as the outcome of a trial, the result of a medical operation, the success of a business venture, or the winner of a football game. Such assessments play an important role in deciding, respectively, whether to go to court, undergo surgery, invest in the venture, or bet on the home team. Uncertainty is usually expressed in verbal terms (e.g., unlikely or probable), but numerical estimates are also common. Weather forecasters, for example, often report the probability of rain (Murphy, 1985), and economists are sometimes required to estimate the chances of recession (Zarnowitz, 1985). The theoretical and practical significance of subjective probability has inspired psychologists, philosophers, and statisticians to investigate this notion from both descriptive and prescriptive standpoints.

Artificial Intelligence has emerged and been considerably developed in the last fifty years together with the advent of the computer age, as a new scientific area aiming at processing information in agreement with the way humans do. Inevitably, such a range of efforts, both theoretical and application-oriented, and motivated by new concerns, has led to modify and enlarge the way basic notions such as uncertainty, belief, knowledge, or evidence could be thought, represented and handled in practice. In particular, there has been a major trend in uncertainty (more specifically, partial belief) modelling, emphasizing the idea that the degree of confidence in an event is not totally determined by the confidence in the opposite event, as assumed in probability theory. Possibility theory (Dubois and Prade, 1988) belongs to this trend that describes partial belief in terms of certainty and plausibility, viewed as distinct concepts. Belief and plausibility functions (Shafer, 1976), or lower and upper probabilities (Walley, 1991) are other important representatives of this trend. The distinctive features of possibility theory are its computational simplicity, and its position as a bridge between numerical and symbolic theories of partial belief for practical reasoning. Possibility theory, based on a non-additive setting that contrasts with probability theory, provides a potentially more qualitative treatment of partial belief, since the operations “max” and “min” play a role somewhat analogous to the sum and the product in probability calculus. There are two different kinds of possibility theory: one is qualitative and the other is quantitative. They share the same kind of set-functions, but they differ when it comes to conditioning and combination

Risk-averse stochastic path detection

Recently, Adler et al. (1984) studied the population dynamics of the meadow jumping mouse, Zapus hudsonius, in Massachusetts. As is typical of contemporary capture-recapture studies on small mammals, they used enumeration estimates of population size and survival despite the known biases of these estimators (Jolly and Dickson, 1983; Nichols and Pollock, 1983). Adler et al. (1984:390) noted that of density and survival rate during the active season for Zapus hudsonius in our study population were considerably lower than those obtained by Nichols and Conley (1982). They also compared estimates of 7 demographic for 3 species trapped on their grids: Z. hudsonius, Peromyscus leucopus, and Microtus pennsylvanicus. They found substantial differences between estimates for Z. hudsonius and the other 2 species. For example, the parameter they termed overall survival (14-day rate) was estimated at 0.13 and 0.09 for Zapus males and females, respectively, and 0.73 and 0.78 for Microtus males and females (Adler et al., 1984:389). On our Michigan study area, W. Conley and I found very similar point estimates of weekly survival probability for these 2 species (0.84 for Zapus, 0.89 for Microtus; Nichols and Conley, 1982), and these estimates did not differ significantly. In this note I do not argue that the apparent interspecific differences reported by Adler et al. (1984) are nonexistent. It may well be that such differences did exist on their study area. Instead, I point out that enumeration estimates are especially inappropriate for such interspecific comparisons and that they cannot be used to draw reasonable inferences about the underlying parameters of interest. This inappropriateness results largely from the fact that enumeration estimators estimate complicated functions of survival and capture probabilities (Nichols and Pollock, 1983) and that different species are often likely to have different capture probabilities. The relevance of capture probability to this argument leads to a second point, which involves the inappropriateness of an estimator for trappability that appears in Adler et al. (1984) and throughout the small mammal literature. The expected value of the enumeration survival rate estimator for capture period i, includes the survival and capture probabilities for every capture period subsequent to i in the entire experiment (see Nichols and Pollock, 1983: eq. 6). Because of this dependence on capture probability, 2 studied populations with identical survival probabilities but different capture probabilities would be expected to yield different enumeration survival estimates. Numerical examples of this problem appear in Figs. 1D and 1E of Nichols and Pollock (1983). These 2 examples show enumeration survival estimates for a single population with constant true survival over 12 capture periods. The difference between the survival estimates for periods with high (0.7) and low (0.3) capture probabilities exemplifies the problem that exists with interspecific comparisons in which the species do not have equal capture probabilities. Even if true survival probabilities are equal, the species with the lower capture probabilities will have lower enumeration survival estimates. W. Conley and I found substantial differences in the Jolly-Seber capture probability estimates for Z. hudsonius and M. pennsylvanicus trapped at the same times on the same grid (Table 1). Estimates for Zapus were consistently lower than those for Microtus until mid-September. We speculated that the crimped oats bait may have been especially attractive during the period of rapid weight gain preceding hibernation, resulting in the high Zapus capture probabilities in September. In any case, these differences in capture probability would have caused problems if we had used enumeration estimates to compare these 2 species. Enumeration estimates of population size and survival rate would have been biased low for both species, but the absolute value of the relative bias would have been greater for Zapus because of the lower capture probabilities. Thus, even if survival rate was similar for the 2 species, as was the case in our study, the enumeration estimate would have been lower for Zapus. Five of the 7 parameter estimates used in the interspecific comparisons of Adler et al. (1984) are functions of capture probability. Any differences between estimates for the different species may then reflect either true differences in the parameter of interest or differences in capture probability. For 3 of these estimators, average density, overall survival, and transiency rate, the effects of differences in capture probability 590 Vol. 67, No. 3

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities. ... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.