What sort of evidence can confer the strongest support to a hypothesis? A natural answer is that the evidence entails the hypothesis. Roush claims that the likelihood ratio measure of degree of incremental support can deliver this intuitively natural result, and regards it as unifying “[the] account of induction and deduction in the only way that makes sense”. In this paper, we highlight a difficulty in the treatment of this case, and question the great significance that is attached (...) to this measure and its alleged capacity to accommodate the logicality requirement. We contrast the likelihood ratio measure with other measures, and argue that problems still emerge in light of tensions with plausible requirements for confirmation measures more generally. (shrink)

What sort of evidence can confer the strongest support to a hypothesis? A natural answer is that the evidence entails the hypothesis. Roush (Tracking Truth: Knowledge, Evidence, and Science, Clarendon Press, Oxford, 2005) claims that the likelihood ratio measure of degree of incremental support can deliver this intuitively natural result, and regards it as unifying “[the] account of induction and deduction in the only way that makes sense” (p. 163). In this paper, we highlight a difficulty in the (...) treatment of this case, and question the great significance that is attached to this measure and its alleged capacity to accommodate the logicality requirement. We contrast the likelihood ratio measure with other measures (such as the Kemeny–Oppenheim measure and the difference measure), and argue that problems still emerge in light of tensions with plausible requirements for confirmation measures more generally. (shrink)

In the recent literature on confirmation there are two leading approaches to the provision of a probabilistic measure of the degree to which a hypothesis is confirmed by evidence. The first is to construe the degree to which evidence E confirms hypothesis H as a function that is directly proportional to p and inversely proportional to p . I shall refer to this as the probability approach. The second approach construes the notion as a function that is directly proportional to (...) the true-positive rate – the probability of the evidence if the hypothesis is true, p – and inversely proportional to the false-positive rate – the probability of the evidence if the hypothesis is false, p . These reverse conditional probabilities – of the evidence on the truth or falsehood of the hypothesis – are sometimes known as likelihoods. I shall refer to the approach to confirmation that uses them as the likelihood approach.For each of these approaches, there are two obvious options as to how to define the confirmation function. The first is to define it as the difference between the magnitude that is treated as increasing confirmation or p ) and the magnitude that is treated as decreasing confirmation or p ). The second is to define confirmation as the quotient of these two magnitudes.This yields four different measures of confirmation, represented in the following table: Differences RatiosProbabilities PD = p – p1 PR = Graphic2Likelihoods LD = p – p3 LR = Graphic4All these measures agree on whether or not E provides some support for H. Intuitively, we want to say that E provides some support for H just in case learning that E is true would make us assign a higher probability to H, i.e. when p(H …. (shrink)

We propose a new hybrid data mining method for predicting protein-protein interactions combining Likelihood-Ratio with rule induction algorithms. In essence, the new method consists of using a rule induction algorithm to discover rules representing partitions of the data, and then the discovered rules are interpreted as “bins” which are used to compute likelihood ratios. This new method is applied to the prediction of protein-protein interactions in the Saccharomyces Cerevisiae genome, using predictive genomic features in an integrated scheme. (...) The results show that the new hybrid method outperforms a pure likelihood ratio based approach. (shrink)

This paper presents a new argument for the likelihood ratio measure of confirmation by showing that one of the adequacy criteria used in another argument can be replaced by a more plausible and better supported criterion which is a special case of the weak likelihood principle. This new argument is also used to show that the likelihood ratio measure is to be preferred to a measure that has recently received support in the literature.

Probability ratio and likelihood ratio measures of inductive support and related notions have appeared as theoretical tools for probabilistic approaches in the philosophy of science, the psychology of reasoning, and artificial intelligence. In an effort of conceptual clarification, several authors have pursued axiomatic foundations for these two families of measures. Such results have been criticized, however, as relying on unduly demanding or poorly motivated mathematical assumptions. We provide two novel theorems showing that probability ratio and (...) class='Hi'>likelihood ratio measures can be axiomatized in a way that overcomes these difficulties. (shrink)

Bayesian confirmation theory is rife with confirmation measures. Zalabardo focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio (...) measure but not by any of the other three measures. Glass and McCartney, hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p and a certain other probability involving H. I then evaluate G&M’s argument in light of them. (shrink)

There are different Bayesian measures to calculate the degree of confirmation of a hypothesis H in respect of a particular piece of evidence E. Zalabardo (Analysis 69:630–635, 2009) is a recent attempt to defend the likelihood-ratio measure (LR) against the probability-ratio measure (PR). The main disagreement between LR and PR concerns their sensitivity to prior probabilities. Zalabardo invokes intuitive plausibility as the appropriate criterion for choosing between them. Furthermore, he claims that it favours the ordering of pairs (...) evidence/hypothesis generated by LR. We will argue, however, that the intuitive non-numerical example provided by Zalabardo does not show that prior probabilities do not affect the degree of confirmation. On account of this, we conclude that there is no compelling reason to endorse LR qua measure of degree of confirmation. On the other side, we should not forget some technicalities which still benefit PR. (shrink)

When the likelihood ratio approach is employed for evidential reasoning in law, it is often necessary to employ subjective probabilities, which are probabilities derived from the opinions and judgement of a human. At least three concerns arise from the use of subjective probabilities in legal applications. Firstly, human beliefs concerning probabilities can be vague, ambiguous and inaccurate. Secondly, the impact of this vagueness, ambiguity and inaccuracy on the outcome of a probabilistic analysis is not necessarily fully understood. Thirdly, (...) the provenance of subjective probabilities and the associated potential sources of vagueness, ambiguity and inaccuracy tend to be poorly understood, making it difficult for the outcome of probabilistic reasoning to be explained and validated, which is crucial in legal applications. The former two concerns have been addressed by a wide body of research in AI. The latter, however, has received little attention. This paper presents a novel approach to employ argumentation to reason about probability distributions in probabilistic models. It introduces a range of argumentation schemes and corresponding sets of critical questions for the construction and validation of argument models that define sets of probability distributions. By means of an extended example, the paper demonstrates how the approach, argumentation schemes and critical questions can be employed for the development of models and their validation in legal applications of the likelihood ratio approach to evidential reasoning. (shrink)

I shall offer some very plausible assumptions for the measure of confirmation and show that they imply that E confirms H relative to background K to degree f/PK), where f is a strictly increasing function. An additional assumption about how measures of confirmation combine then makes f be proportional to a logarithm.

Likelihoodism is the view that the degree of evidential support should be analysed and measured in terms of likelihoods alone. The paper considers and responds to a popular criticism that a likelihoodist framework is too restrictive to guide belief. First, I show that the most detailed and rigorous version of this criticism, as put forward by Gandenberger (2016), is unsuccessful. Second, I provide a positive argument that a broadly likelihoodist framework can accommodate guidance for comparative belief, even when objectively well-grounded (...) prior probabilities are not available. As I show, the shift from non-relational to comparative probabilities opens up a new space for addressing the belief guidance problem for likelihoodism. (shrink)

This article shows that a slight variation of the argument in Milne 1996 yields the log‐likelihood ratio l rather than the log‐ratio measure r as “the one true measure of confirmation. ” *Received December 2006; revised December 2007. †To contact the author, please write to: Formal Epistemology Research Group, Zukunftskolleg and Department of Philosophy, University of Konstanz, P.O. Box X906, 78457 Konstanz, Germany; e‐mail: franz.huber@uni‐konstanz.de.

Female waist-to-hip ratio (WHR) has generally been an important general predictor of ratings of physical attractiveness and related characteristics. Individual differences in ratings do exist, however, and may be related to differences in the reproductive tactics of the male raters such as pursuit of short-term or long-term relationships and adjustments based on perceptions of one’s own quality as a mate. Forty males, categorized according to sociosexual orientation and physical qualities (WHR, Body Mass Index, and self-rated desirability), rated female models (...) on both attractiveness and likelihood they would approach them. Sociosexually restricted males were less likely to approach females rated as most attractive (with 0.68–0.72 WHR), as compared with unrestricted males. Males with lower scores in terms of physical qualities gave ratings indicating more favorable evaluations of female models with lower WHR. The results indicate that attractiveness and willingness to approach are overlapping but distinguishable constructs, both of which are influenced by variations in characteristics of the raters. (shrink)

College students from small, medium, and large institutions with either a modified or no honor code were presented with cheating scenarios and asked to rate how dishonest they perceived the behavior to be and the likelihood that they would report it. No main effects were found for institution size or type of honor code. Student–faculty ratio was not correlated with responses to the cheating scenarios. Students from modified honor code schools perceived more severe punishments for cheating and understood (...) the reporting process better than students from non–honor code schools. Implications for modified honor code systems are discussed. (shrink)

BackgroundDepression is a serious mental disorder often accompanied by emotional and physiological disorders. Visceral fat index is the current standard method in the evaluation of visceral fat deposition. In this study, we explored the association between VAI and depression in the American population using NHANES data.MethodsA total of 2,577 patients were enrolled for this study. Data were collected through structured questionnaires. Subgroup analysis for the relationship between VAI and depression was evaluated using multivariate regression analysis after adjustment for potential confounding (...) factors.ResultsFor every 1 unit increase in VAI, the clinical depression increased by 14%. High VAI scores increased the highest risk of developing depression. Subgroup analysis demonstrated a strong and stable association between VAI and the development of depression.ConclusionOur study showed that depressive symptoms are associated with a high ratio of visceral adiposity index after controlling confounding factors. (shrink)

It is well known that Bayes’ theorem (with likelihood ratios) can be used to calculate the impact of evidence, such as a ‘match’ of some feature of a person. Typically the feature of interest is the DNA profile, but the method applies in principle to any feature of a person or object, including not just DNA, fingerprints, or footprints, but also more basic features such as skin colour, height, hair colour or even name. Notwithstanding concerns about the extensiveness of (...) databases of such features, a serious challenge to the use of Bayes in such legal contexts is that its standard formulaic representations are not readily understandable to non-statisticians. Attempts to get round this problem usually involve representations based around some variation of an event tree. While this approach works well in explaining the most trivial instance of Bayes’ theorem (involving a single hypothesis and a single piece of evidence) it does not scale up to realistic situations. In particular, even with a single piece of match evidence, if we wish to incorporate the possibility that there are potential errors (both false positives and false negatives) introduced at any stage in the investigative process, matters become very complex. As a result we have observed expert witnesses (in different areas of speciality) routinely ignore the possibility of errors when presenting their evidence. To counter this, we produce what we believe is the first full probabilistic solution of the simple case of generic match evidence incorporating both classes of testing errors. Unfortunately, the resultant event tree solution is too complex for intuitive comprehension. And, crucially, the event tree also fails to represent the causal information that underpins the argument. In contrast, we also present a simple-to-construct graphical Bayesian Network (BN) solution that automatically performs the calculations and may also be intuitively simpler to understand. Although there have been multiple previous applications of BNs for analysing forensic evidence—including very detailed models for the DNA matching problem, these models have not widely penetrated the expert witness community. Nor have they addressed the basic generic match problem incorporating the two types of testing error. Hence we believe our basic BN solution provides an important mechanism for convincing experts—and eventually the legal community—that it is possible to rigorously analyse and communicate the full impact of match evidence on a case, in the presence of possible errors. (shrink)

We comment on the contributions of Dahlman and of Fenton et al., who both suggested a Bayesian approach to analyze the Simonshaven case. We argue that analyzing a full case with a Bayesian approach is not feasible, and that there are serious problems with assigning actual numbers to probabilities and priors. We also discuss the nature of Bayesian thinking in court, and the nature and interpretation of the likelihood ratio. In particular, we discuss what it could mean that (...) a likelihood ratio is in some sense uncertain. (shrink)

Reasons transmit. If one has a reason to attain an end, then one has a reason to effect means for that end: reasons are transmitted from end to means. I argue that the likelihood ratio (LR) is a compelling measure of reason transmission from ends to means. The LR measure is superior to other measures, can be used to construct a condition specifying precisely when reasons transmit, and satisfies intuitions regarding end-means reason transmission in a broad array of (...) cases. (shrink)

There is a plethora of confirmation measures in the literature. Zalabardo considers four such measures: PD, PR, LD, and LR. He argues for LR and against each of PD, PR, and LD. First, he argues that PR is the better of the two probability measures. Next, he argues that LR is the better of the two likelihood measures. Finally, he argues that LR is superior to PR. I set aside LD and focus on the trio of PD, PR, and (...) LR. The question I address is whether Zalabardo succeeds in showing that LR is superior to each of PD and PR. I argue that the answer is negative. I also argue, though, that measures such as PD and PR, on one hand, and measures such as LR, on the other hand, are naturally understood as explications of distinct senses of confirmation. (shrink)

According to Bayesian confirmation theory, evidence E (incrementally) confirms (or supports) a hypothesis H (roughly) just in case E and H are positively probabilistically correlated (under an appropriate probability function Pr). There are many logically equivalent ways of saying that E and H are correlated under Pr. Surprisingly, this leads to a plethora of non-equivalent quantitative measures of the degree to which E confirms H (under Pr). In fact, many non-equivalent Bayesian measures of the degree to which E confirms (or (...) supports) H have been proposed and defended in the literature on inductive logic. I provide a thorough historical survey of the various proposals, and a detailed discussion of the philosophical ramifications of the differences between them. I argue that the set of candidate measures can be narrowed drastically by just a few intuitive and simple desiderata. In the end, I provide some novel and compelling reasons to think that the correct measure of degree of evidential support (within a Bayesian framework) is the (log) likelihood ratio. The central analyses of this research have had some useful and interesting byproducts, including: (i ) a new Bayesian account of (confirmationally) independent evidence, which has applications to several important problems in con- firmation theory, including the problem of the (confirmational) value of evidential diversity, and (ii ) novel resolutions of several problems in Bayesian confirmation theory, motivated by the use of the (log) likelihood ratio measure, including a reply to the Popper-Miller critique of probabilistic induction, and a new analysis and resolution of the problem of irrelevant conjunction (a.k.a., the tacking problem). (shrink)

Background and aims: Recent investigations have highlighted the value of neuropsychological testing for the assessment and screening of Alcohol-Related Brain Damage. The aim of the present study was to evaluate the suitability of the Addenbrooke’s Cognitive Examination and the Repeatable Battery for the Assessment of Neuropsychological Status for this purpose. Methods: Comparing 28 participants with ARBD and 30 alcohol-dependent participants without ARBD we calculated Area Under the Curve statistics, sensitivity and specificity values, base-rate adjusted predictive values, and likelihood ratios (...) for both tests. Results: High levels of screening accuracy were found for the total scores of both the ACE-III and RBANS at multiple cut-off points. Removing participants with a history of polysubstance from the samples improved the diagnostic capabilities of the RBANS substantially, while only minor improvements to the ACE-III’s accuracy were observed. Conclusions: Overall, both the ACE-III and RBANS are suitable tools for ARBD screening within an alcohol-dependent population, though the RBANS is the superior of the two. Clinicians using these tools for ARBD screening should be cautious of false-positive outcomes and should therefore combine them with other assessment methods and more detailed neuropsychological testing before reaching diagnostic decisions. (shrink)

The old evidence problem affects any probabilistic confirmation measure based on comparing pr(H/E) and pr(H). The article argues for the following points: (1) measures based on likelihood ratios also suffer old evidence difficulties; (2) the less-discussed synchronic old evidence problem is, in an important sense, the most acute; (3) prominent attempts to solve or dissolve the synchronic problem fail; (4) a little-discussed variant of the standard measure avoids the problem, in an appealing way; and (5) this measure nevertheless reveals (...) a different problem for probabilistic confirmation measures, a problem that is unlikely to lend itself to formal solution. (shrink)

Legal scholarship exploring the nature of evidence and the process of juridical proof has had a complex relationship with formal modeling. As evident in so many fields of knowledge, algorithmic approaches to evidence have the theoretical potential to increase the accuracy of fact finding, a tremendously important goal of the legal system. The hope that knowledge could be formalized within the evidentiary realm generated a spate of articles attempting to put probability theory to this purpose. This literature was both insightful (...) and frustrating. Much light was shed on the legal system, but it also quickly became evident that the tools of probability theory were in many ways ill-constructed for the task. Fundamental incompatibilities between the structure of legal decision making and the extant formal tools were identified, and it became evident that many of the purported explanations of legal phenomena were internally inconsistent. As a consequence, interest in this type of formal modeling declined, and attention was directed toward different kinds of explanations of the phenomena. Perhaps under the influence of a recent trend toward various types of formal modeling in legal scholarship, a recent burst of articles, rather than attempting to explain the macro structure of trials, which was the previous object of interest, attempts to quantify the probative value of various items of evidence in ways consistent with the formal features of various probability theories, and then to study decision making from that perspective. For example, the value of evidence is often purported to be its likelihood ratio, that is, the probability of discovering or receiving the evidence given a hypothesis (e.g., the defendant did it) divided by the probability of discovering or receiving the evidence given the negation of the hypothesis (the defendant didn't do it). Alternatively, the value of evidence is purported (more contextually) to be the information gain it provides, defined as the increase in probability it provides for a hypothesis above the probability of the hypothesis based on the other available evidence. Both conceptions then assume that all of the various probability assessments conform or ought to conform to the dictates of Bayes' theorem (that maintains consistency among such assessments); empirical studies are then done testing the extent to which this is so and proposing how the law can increase the probability that it is so. The general criticisms of using Bayes' theorem as a formal model of juridical proof are well known and were integral to the last wave of interest in formal modeling of the evidentiary process. This paper thus for the most part puts aside that more general issue, and focuses specifically on mathematical modeling of the value of particular items of evidence. The paper demonstrates that formal modeling has only limited value in explaining the value of legal evidence, much more limited than those constructing and discussing the models assume, and thus that the conclusions they draw about the value of evidence are unwarranted. This is done through a discussion of four recent examples that attempt to quantify evidence relating to, respectively, carpet fibers, infidelity, DNA random-match evidence, and character evidence used to impeach a witness. This article thus makes two contributions. First, and most importantly, it is another demonstration of the complex relationship between algorithmic tools and legal decision making. Second, at a minimum it points out serious pitfalls for analytical or empirical studies of juridical proof. (shrink)

Bayesian epistemology suggests various ways of measuring the support that a piece of evidence provides a hypothesis. Such measures are defined in terms of a subjective probability assignment, pr, over propositions entertained by an agent. The most standard measure (where “H” stands for “hypothesis” and “E” stands for “evidence”) is: the difference measure: d(H,E) = pr(H/E) - pr(H).0 This may be called a “positive (probabilistic) relevance measure” of confirmation, since, according to it, a piece of evidence E qualitatively confirms a (...) hypothesis H if and only if pr(H/E) > pr(H), where qualitative disconfirmation is characterized by replacing “>” with “ “ with “=”. Other more or less standard positive relevance measures that have been proposed are: the log-ratio measure: r(H,E) = log[pr(H/E)/pr(H)] and the log-likelihood-ratio measure: l(H,E) = log[pr(E/H)/pr(E/~H)]. (shrink)

Assuming that votes are independent, the epistemically optimal procedure in a binary collective choice problem is known to be a weighted supermajority rule with weights given by personal log-likelihood-ratios. It is shown here that an analogous result holds in a much more general model. Firstly, the result follows from a more basic principle than expected-utility maximisation, namely from an axiom (Epistemic Monotonicity) which requires neither utilities nor prior probabilities of the ‘correctness’ of alternatives. Secondly, a person’s input need not (...) be a vote for an alternative, it may be any type of input, for instance a subjective degree of belief or probability of the correctness of one of the alternatives. The case of a profile of subjective degrees of belief is particularly appealing, since here no parameters such as competence parameters need to be known. (shrink)

The crisis of replication has led many to blame statistical significance tests for making it too easy to find impressive looking effects that do not replicate. However, the very fact it becomes difficult to replicate effects when features of the tests are tied down actually serves to vindicate statistical significance tests. While statistical significance tests, used correctly, serve to bound the probabilities of erroneous interpretations of data, this error control is nullified by data-dredging, multiple testing, and other biasing selection effects. (...) Arguments claiming to vitiate statistical significance tests attack straw person variants of tests that commit well-known fallacies and misinterpretations. There is a tension between popular calls for preregistration – arguably, one of the most promising ways to boost replication – and accounts that downplay error probabilities: Bayes Factors, Bayesian posteriors, likelihood ratios. By underscoring the importance of error control for well testedness, the replication crisis points to reformulating tests so as to avoid fallacies and report the extent of discrepancies that are and are not indicated with severity. (shrink)

– Foundation: Probabilistic Confirmation (c) from a Logical POV ∗ cph, eq as a “relevant” quantitative generalization of pe  hq ∗ cph, eq, so understood, is not Prpe  hq or Prph | eq, etc. ∗ cph, eq is something akin (ordinally) to the likelihood ratio..

We investigate the epistemological consequences of a positive polymerase chain reaction SARS-CoV test for two relevant hypotheses: V is the hypothesis that an individual has been infected with SARS-CoV-2; C is the hypothesis that SARS-CoV-2 is the cause of flu-like symptoms in a given patient. We ask two fundamental epistemological questions regarding each hypothesis: First, how much confirmation does a positive test lend to each hypothesis? Second, how much evidence does a positive test provide for each hypothesis against its negation? (...) We respond to each question within a formal Bayesian framework. We construe degree of confirmation as the difference between the posterior probability of the hypothesis and its prior, and the strength of evidence for a hypothesis against its alternative in terms of their likelihood ratio. We find that test specificity—and coinfection probabilities when making inferences about C—were key determinants of confirmation and evidence. Tests with 8) for V against ¬V regardless of sensitivity. Accordingly, low specificity tests could not provide strong evidence in favor of C in all plausible scenarios modeled. We also show how a positive influenza A test disconfirms C and provides weak evidence against C in dependence on the probability that the patient is influenza A infected given that his/her symptoms are not caused by SARS-CoV-2. Our analysis points out some caveats that should be considered when attributing symptoms or death of a positively tested patient to SARS-CoV-2. (shrink)

In this paper we argue that it is often adaptive to use one's background beliefs when interpreting information that, from a normative point of view, is incomplete. In both of the experiments reported here participants were presented with an item possessing two features and were asked to judge, in the light of some evidence concerning the features, to which of two categories it was more likely that the item belonged. It was found that when participants received evidence relevant to just (...) one of these hypothesised categories they used their background beliefs to interpret this information. In Experiment 2, on the other hand, participants behaved in a broadly Bayesian manner when the evidence they received constituted a completed likelihood ratio. We discuss the circumstances under which participants, when making their judgements, consider the alternative hypothesis. We conclude with a discussion of the implications of our results for an understanding of hypothesis testing, belief revision, and categorisation. (shrink)

This paper proposes a uniform method for constructing tests, confidence regions and point estimates which is called exact since it reduces to Fisher's so-called exact test in the case of the hypothesis of independence in a 2 × 2 contingency table. All the wellknown standard tests based on exact sampling distributions are instances of the exact test in its general form. The likelihood ratio and x2 tests as well as the maximum likelihood estimate appears as asymptotic approximations (...) to the corresponding exact procedures. (shrink)

Much forensic inference based upon DNA evidence is made assuming Hardy-Weinberg Equilibrium (HWE) for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, and their limitations become more obvious when testing for deviation within multiallelic DNA loci. The most popular methods-Chi-square and Likelihood-ratio tests-are based on asymptotic results and cannot guarantee a good performance in the presence of low frequency genotypes. Since the parameter space dimension increases at a quadratic (...) rate on the number of alleles, some authors suggest applying sequential methods, where the multiallelic case is reformulated as a sequence of “biallelic” tests. However, in this approach it is not obvious how to assess the general evidence of the original hypothesis; nor is it clear how to establish the significance level for its acceptance/rejection. In this work, we introduce a straightforward method for the multiallelic HWE test, which overcomes the aforementioned issues of sequential methods. The core theory for the proposed method is given by the Full Bayesian Significance Test (FBST), an intuitive Bayesian approach which does not assign positive probabilities to zero measure sets when testing sharp hypotheses. We compare FBST performance to Chi-square, Likelihood-ratio and Markov chain tests, in three numerical experiments. The results suggest that FBST is a robust and high performance method for the HWE test, even in the presence of several alleles and small sample sizes. (shrink)

The main point of the paper is to show how popular probabilistic measures of incremental confirmation and statistical relevance with qualitatively different features can be embedded smoothly in generalized parametric families. In particular, I will show that the probability difference, log probability ratio, log likelihood ratio, odds difference, so-called improbability difference, and Gaifman’s measures of confirmation can all be subsumed within a convenient biparametric continuum. One intermediate step of this project may have interest on its own, as (...) it provides a unified representation of graded belief of which both probabilities and odds are special cases. (shrink)

In the debate about the legal value of naked statistical evidence, Di Bello argues that (1) the likelihood ratio of such evidence is unknown, (2) the decision-theoretic considerations indicate that a conviction based on such evidence is unacceptable when expected utility maximization is combined with fairness constraints, and (3) the risk of mistaken conviction based on such evidence cannot be evaluated and is potentially too high. We argue that Di Bello’s argument for (1) works in a rather narrow (...) context, and that (1) is not exactly in line with the way expert witnesses are required to formulate their opinions. Consequently, Di Bello’s argument for (2), which assumes (1), does not apply uniformly to all convictions based on naked evidence. Moreover, if Di Bello’s analysis is correct, it applies also to eyewitness testimony, given empirical results about its quality, and so the distinctions drawn by DiBello cut across the distinction between naked statistical evidence and other types of evidence. Finally, if we weaken the rather strong requirement of precise measurability of the risk of mistaken conviction, to the availability of reasonable but imprecise and fallible estimates, many field and empirical studies show that often the risk of mistaken conviction based on naked statistical evidence can be estimated to a similar extent as the risk of mistaken conviction based on any other sort of evidence. (shrink)

Think of confirmation in the context of the Ravens Paradox this way. The likelihood ratio measure of incremental confirmation gives us, for an observed Black Raven and for an observed non-Black non-Raven, respectively, the following “full” likelihood ratios.

Over the last decades, understanding the sources of the difficulty of Bayesian problem solving has been an important research goal, with the effects of numerical format and individual numeracy being widely studied. However, the focus on the comprehension of probability numbers has overshadowed the relational reasoning demand of the Bayesian task. This is particularly the case when the statistical data are verbally described since the requested quantitative relation (posterior ratio) is misaligned with the presented ones (prior and likelihood (...) ratios). In this regard, here I develop the proposal that research on Bayesian reasoning might improve by considering the notational alignment framework of mathematical problem-solving. Specifically, this framework can help to understand the sources of the main difficulties underlying Bayesian inferences based on verbal descriptions. In essence, the present proposal supports the general claim in math education regarding the need to foster relational comprehension to avoid misleading alignments and improve problem solving. (shrink)

The Generalized Poisson Distribution (GPD) adds an extra parameter to the usual Poisson distribution. This parameter induces a loss of homogeneity in the stochastic processes modeled by the distribution. Thus, the generalized distribution becomes an useful model for counting processes where the occurrence of events is not homogeneous. This model creates the need for an inferential procedure, to test for the value of this extra parameter. The FBST (Full Bayesian Significance Test) is a Bayesian hypotheses test procedure, capable of providing (...) an evidence measure on sharp hypotheses (where the dimension of the parametric space under the null hypotheses is smaller than that of the full parametric space). The goal of this work is study the empirical properties of the FBST for testing the nullity of extra parameter of the generalized Poisson distribution. Numerical experiments show a better performance of FBST with respect to the classical likelihood ratio test, and suggest that FBST is an efficient and robust tool for this application. (shrink)

Road traffic carnages are global concerns and seemingly on the rise in Ghana. Several risk factors have been studied as associated with road traffic fatalities. However, inadequate road traffic fatality data and inconsistent probability outcomes for RTF remain major challenges. The objective of this study was to illustrate and estimate probability models that can predict road traffic fatalities. We relied on 66,159 recorded casualties who were involved in road traffic accidents in Ghana from 2015 to 2019. Three generalized linear models, (...) namely, logistic regression, probit regression, and linear probability model, were used for the analysis. We found that gender and age groups have significant effects in predicting the probability of road traffic fatality for all three models. Through a likelihood ratio test, however, it was determined that the logit regression model produced consistent probabilities of traffic fatalities which are very close to the actual probability values across the age groups and gender, compared to the other two models. Thus, we recommend intensified campaign for the use of seat belts in vehicles, targeted at the aged and male users of road transport, to reduce the possibility of death in any RTA. (shrink)

In clinical studies, it is important to investigate the effectiveness of different therapeutic designs, especially, multiple treatment groups to one control group. The paper mainly studies homogeneity test of many-to-one risk differences from correlated binary data under optimal algorithms. Under Donner’s model, several algorithms are compared in order to obtain global and constrained MLEs in terms of accuracy and efficiency. Further, likelihood ratio, score, and Wald-type statistics are proposed to test whether many-to-one risk differences are equal based on (...) optimal algorithms. Monte Carlo simulations show the performance of these algorithms through the total averaged estimation error, SD, MSE, and convergence rate. Score statistic is more robust and has satisfactory power. Two real examples are given to illustrate our proposed methods. (shrink)

Three experiments investigated the effect of rarity on people's selection and interpretation of data in a variant of the pseudodiagnosticity task. For familiar (Experiment 1) but not for arbitrary (Experiment 3) materials, participants were more likely to select evidence so as to complete a likelihood ratio when the initial evidence they received was a single likelihood concerning a rare feature. This rarity effect with familiar materials was replicated in Experiment 2 where it was shown that participants were (...) relatively insensitive to explicit manipulations of the likely diagnosticity of rare evidence. In contrast to the effects for data selection, there was an effect of rarity on confidence ratings after receipt of a single likelihood for arbitrary but not for familiar materials. It is suggested that selecting diagnostic evidence necessitates explicit consideration of the alternative hypothesis and that consideration of the possible consequences of the evidence for the alternative weakens the rarity effect in confidence ratings. Paradoxically, although rarity effects in evidence selection and confidence ratings are in the spirit of Bayesian reasoning, the effect on confidence ratings appears to rely on participants thinking less about the alternative hypothesis. (shrink)

We explore the question of whether cost-free uncertain evidence is worth waiting for in advance of making a decision. A classical result in Bayesian decision theory, known as the value of evidence theorem, says that, under certain conditions, when you update your credences by conditionalizing on some cost-free and certain evidence, the subjective expected utility of obtaining this evidence is never less than the subjective expected utility of not obtaining it. We extend this result to a type of update method, (...) a variant of Judea Pearl’s virtual conditionalization, where uncertain evidence is represented as a set of likelihood ratios. Moreover, we argue that focusing on this method rather than on the widely accepted Jeffrey conditionalization enables us to show that, under a fairly plausible assumption, gathering uncertain evidence not only maximizes expected pragmatic utility, but also minimizes expected epistemic disutility. (shrink)

An Intuitive Explanation of Bayesian Reasoning. You should easily recognize, and intuitively understand, the concepts "prior probability", "posterior probability", "likelihood ratio", and "odds ratio". This essay is intended as a sequel to the Intuitive Explanation, but you might skip that introduction if you are already thoroughly Bayesian. Where the Intuitive Explanation focused on providing a firm grasp of Bayesian basics, the Technical Explanation builds, on a Bayesian foundation, theses about human rationality and philosophy of science.

The relationship between corporate social responsibility (CSR) and leadership has received considerable research attention in recent decades. While there have been several qualitative reviews, quantitative and systematic reviews of CSR–leadership links remain absent. The current paper seeks to address this gap by using a bibliometric method to analyze and visualize the evolution and research trends within the CSR–leadership domain. Drawing from a sample of 1432 peer-reviewed articles, we map the landscape of the CSR–leadership research domain and identify key developments and (...) patterns over the period 1994–2020. In addition to highlighting a range of publication trends including key theoretical influences, the findings also reveal an acceleration in the volume and breadth of CSR–leadership research. Seven subdomains (or clusters) are further identified (i.e., board characteristics, responsible leadership, emerging country context, team efficacy, CEO pay fairness, shareholder wealth, and cross-sector social partnership) and discussed based on the keywords using the log-likelihood ratio clustering algorithm. The analysis is further divided into three stages: namely, the initial stage (1991–2003), the rapid development stage (2004–2011), and the maturation stage (2012–present). These stages are used to present more fine-grained insights into the research patterns over time. We conclude by discussing implications of the bibliometric analysis and presenting several opportunities for future research. (shrink)