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The probability heuristics model of syllogistic reasoning.

A probability heuristic model (PHM) for syllogistic reasoning is proposed. An informational ordering over quantified statements suggests simple probability based heuristics for syllogistic reasoning. The most important is the "min-heuristic": choose the type of the least informative premise as the type of the conclusion. The rationality of this heuristic is confirmed by an analysis of the probabilistic validity of syllogistic reasoning which treats logical inference as a limiting case of probabilistic inference. A meta-analysis of past experiments reveals close fits with PHM. PHM also compares favorably with alternative accounts, including mental logics, mental models, and deduction as verbal reasoning. Crucially, PHM extends naturally to generalized quantifiers, such as Most and Few, which have not been characterized logically and are, consequently, beyond the scope of current mental logic and mental model theories. Two experiments confirm the novel predictions of PHM when generalized quantifiers are used in syllogistic arguments. PHM suggests that syllogistic reasoning performance may be determined by simple but rational informational strategies justified by probability theory rather than by logic.

Algorithms↗

Subjective probability of disjunctive hypotheses: local-weight models for decomposition of evidential support.

When the probability of a single member of a set of mutually exclusive and exhaustive possibilities is judged, its alternatives are evaluated as a composite "residual" hypothesis. Support theory (Rottenstreich & Tversky, 1997; Tversky & Koehler, 1994) implies that the process of packing alternatives together in the residual reduces the perceived evidential support for the set of alternatives and consequently inflates the judged probability of the focal hypothesis. Previous work has investigated the global weights that determine the extent to which the overall evidential support for the alternatives is discounted by this packing operation (Koehler, Brenner, & Tversky, 1997). In the present investigation, we analyze this issue in greater detail, examining the local weights that measure the specific contribution of each component hypothesis included implicitly in the residual. We describe a procedure for estimating local weights and introduce a set of plausible properties that impose systematic ordinal relationships among local weights. Results from four experiments testing these properties are reported, and a local-weight model is developed that accounts for nearly all of the variance in the probability judgments in these empirical tests. Local weights appear to be sensitive both to the individual component with which they are associated and to the residual hypothesis in which the component resides.

Cognition↗

Frequency, probability, and prediction: easy solutions to cognitive illusions?

Many errors in probabilistic judgment have been attributed to people's inability to think in statistical terms when faced with information about a single case. Prior theoretical analyses and empirical results imply that the errors associated with case-specific reasoning may be reduced when people make frequentistic predictions about a set of cases. In studies of three previously identified cognitive biases, we find that frequency-based predictions are different from-but no better than-case-specific judgments of probability. First, in studies of the "planning fallacy, " we compare the accuracy of aggregate frequency and case-specific probability judgments in predictions of students' real-life projects. When aggregate and single-case predictions are collected from different respondents, there is little difference between the two: Both are overly optimistic and show little predictive validity. However, in within-subject comparisons, the aggregate judgments are significantly more conservative than the single-case predictions, though still optimistically biased. Results from studies of overconfidence in general knowledge and base rate neglect in categorical prediction underline a general conclusion. Frequentistic predictions made for sets of events are no more statistically sophisticated, nor more accurate, than predictions made for individual events using subjective probability.

Bayes Theorem↗

Stochastic models of tumor growth and the probability of elimination by cytotoxic cells.

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is T' = gamma 0T - k'6T/(K1 + T) where gamma 0, k'6 and K1 are positive constants. The parameter of most import is lambda 0 = gamma 0 - k'6/K1 which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case lambda = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.

Cell Survival↗

On the survival probability of a slightly advantageous mutant gene in a multitype population: a multidimensional branching process model.

The estimated survival probability of a slightly supercritical Galton-Watson process is generalized to a multitype branching process. The result is used to estimate the probability of initial success of a mutant gene whose effect on the individual carrier depends on the carrier's sex, class, etc. The probability of initial success is also estimated in a case where the effect of the mutation is manifested in terms of the distribution of types within one's progeny, e.g. in a case of a change in the sex ratio.

Genes↗

Visual contrast detection by a single channel versus probability summation among channels.

It is now generally accepted that the human visual system consists of subsystems ("channels") that may be activated in parallel. According to some models of detection, detection is by probability summation among channels, while in other models it is assumed that detection is by a single channel that may even be tuned specifically to the stimulus pattern (detection by a matched filter). So far, arguments in particular for the hypothesis of probability summation are based on plausibility considerations and on demonstrations that the data from certain detection experiments are compatible with this hypothesis. In this paper it is shown that linear contrast interrelationship functions together with a property of a large class of distribution functions (strict log-concavity or log-convexity on the relevant set of contrasts/intensities) uniquely point to detection by a single channel. In particular, models of detection by probability summation based on Quick's Model are incompatible with linear contrast interrelationship functions. Sufficient (and observable) conditions for the strict log-concavity/log-convexity of distribution functions are presented.

Humans↗

The influence of the probability of random performance of a reaction to be developed on the rate of formation of an instrumental reflex.

An instrumental defense reflex of pedal pressing during the action of a conditional signal (light) was developed in rats. The process of learning given a varying probability of the random performance of the instrumental reaction, p(a), which depends in particular on the duration of the intervals between the action of the conditional stimulus, was studied. It was shown that the rate of formation of a conditioned reflex is higher for p(a) = 0.25 than for p(a) = 0.05. The rate of formation of the conditioned reflex also depends on the frequency of the positive reinforcement of the correct reactions (from 50 to 100%). With a high p(a), the rate of formation of the reflex in question does not depend on the frequency of reinforcement of the positive responses; a feedback relationship is observed in the case of a low p(a). The hypothesis is advanced that not only does the behavior itself change the probability of contact with the external object but the original (a priori) probability substantially influences the formation of the behavior as well.

Animals↗

Solution of a gene divergence problem under arbitrary stable nucleotide transition probabilities.

A nucleic acid chain L nucleotides in length, with the specific base sequence B1B2....BL, each Bi being A, G, C, or T, is defined by the L-dimensional vector B = (B1, B2, ..., BL), the kth position in the chain being occupied by the base Bk. Let pBB, be the twelve given constant nonnegative transition probabilities that in a specified position the base B is replaced by the base B' in a single step, and let P(X)BB, be the probability that the position goes from base B to B' in X steps. An exact analytical expression for P(X)BB' is derived. Assuming that each base mutates independently of the others, an exact expression is derived for the probability P(X)BB' that the initial gene sequence B goes to a sequence B' = (B'1, B'2; ..., B'L) after X = (X1, X2, ..., XL) base replacements, where Xk is the number of single step base replacements in the kth position. The resulting equations allow a more precise accounting for the effects of Darwinian natural selection in molecular evolution than does the idealized but biologically less accurate assumption that each of the four nucleotides is equally likely to mutate to and be fixed as one of the other three. Illustrative applications of the theory to some problems in biological evolution are given.

Base Sequence↗

Estimate of the sticking probability for cells in uniform shear flow with adhesion caused by specific bonds.

A theory is developed for the aggregation rate of cells in uniform shear flow when the cell-cell adhesion is mediated by bonds between specific molecules on the cell surfaces such as antigen and antibody or lectin and carbohydrate. The theory is based on estimates of the frequency and duration of cell-cell collisions and of the number of bonds formed and required to hold the cells together. For high shear rates, the sticking probability is a function of a single dimensionless parameter, A, that is proportional to G-2, with G the shear rate. For low shear rates, the sticking probability is a function of a second dimensionless parameter, A' proportional to G-1. In either case the rate of cell-cell sticking is a maximum when A (or A') congruent to 1.0. For small values of A (or A') the cells collide frequently, but do not stick, whereas for large values of A (or A') the cells collide infrequently, but stick with larger probability. Studies in Couette viscometer or other flow having approximately uniform shear can test these models.

Animals↗

Probability prediction in multistate survival models for patients with chronic myeloid leukaemia.

In order to find an appropriate model suitable for a multistate survival experiment, 634 patients with chronic myeloid leukaemia (CML) were selected to illustrate the method of analysis. After transplantation, there were 4 possible situations for a patient: disease free, relapse but still alive, death before relapse, and death after relapse. The last 3 events were considered as treatment failure. The results showed that the risk of death before relapse was higher than that of the relapse, especially in the first year after transplantation with competing-risk method. The result of patients with relapse time less than 12 months was much poor by the Kaplan-Meier method. And the multistate survival models were developed, which were detailed and informative based on the analysis of competing risks and Kaplan-Meier analysis. With the multistate survival models, a further analysis on conditional probability was made for patients who were disease free and still alive at month 12 after transplantation. It was concluded that it was possible for an individual patient to predict the 4 possible probabilities at any time. Also the prognoses for relapse either death or not and death either before or after relapse may be given. Furthermore, the conditional probabilities for patients who were disease free and still alive in a given time after transplantation can be predicted.

Bone Marrow Transplantation↗

Comparison of absolute thresholds derived from an adaptive forced-choice procedure and from reaction probabilities and reaction times in a simple reaction time paradigm.

An understanding of the auditory system's operation requires knowledge of the mechanisms underlying thresholds. In this work we compare detection thresholds obtained with a three-interval-three-alternative forced-choice paradigm with reaction thresholds extracted from both reaction probabilities (RP) and reaction times (RT) in a simple RT paradigm from the same listeners under otherwise nearly identical experimental conditions. Detection thresholds, RP, and RT to auditory stimuli exhibited substantial variation from session to session. Most of the intersession variation in RP and RT could be accounted for by intersession variation in a listener's absolute sensitivity. The reaction thresholds extracted from RP were very similar, if not identical, to those extracted from RT. On the other hand, reaction thresholds were always higher than detection thresholds. The difference between the two thresholds can be considered as the additional amount of evidence required by each listener to react to a stimulus in an unforced design on top of that necessary for detection in the forced-choice design. This difference is inversely related to the listener's probability of producing false alarms. We found that RT, once corrected for some irreducible minimum RT, reflects the time at which a given stimulus reaches the listener's reaction threshold. This suggests that the relationships between simple RT and loudness (reported in the literature) are probably caused by a tight relationship between temporal summation at threshold and temporal summation of loudness.

Adult↗

Subjective probability of accident and apprehension in relation to self-other bias, age, and reported behavior.

Responses from over 900 male drivers were examined on items measuring the subjective probability of two adverse driving scenarios (accident and apprehension) for a number of speeding and drink-driving-related events. The items were constructed to focus on self versus average referent driver comparisons. Respondents were categorised into "offenders" and "non-offenders" according to their reported behaviour concerning speeding and drink-driving. The data were analysed using repeated measures analysis of variance. The results indicated that both accident and apprehension are seen as less probable when self is the reference compared to the average driver. There was evidence to suggest that this self-favouring bias did vary over presented scenarios. It was suggested that such perceptions may vary in relation to the perceived influence of driver skill within scenarios. Weak age effects were found, but it was concluded that, generally, self-favouring biases were common across all age groups. Concerning the offender/nonoffender classification, the results indicated that speeders and drink-drivers generally perceived the probability of apprehension and accident as being lower than did the nonoffender groups. There was no evidence to suggest that self-average driver biases were associated with offender/nonoffender grouping.

Accidents, Traffic↗

Comparison of the multivariate analysis and CADENZA systems for determination of the probability of coronary artery disease.

The accuracy of 2 discriminate systems for diagnosis of coronary artery disease (CAD), multivariate analysis (MVA) and Bayesian analysis (CADENZA), was evaluated in 113 patients undergoing electrocardiographic stress testing and coronary angiography. MVA uses weighting factors (F values) generated from our patient data, whereas CADENZA uses probabilities gleaned from an extensive review of the American literature. Overall accuracy was similar. MVA had a higher sensitivity for 1-vessel CAD (75 versus 33%), but CADENZA was better for determining the severity of CAD. The 2 systems provided posterior probabilities for disease that were highly correlated (r = 0.56; p less than 0.001). Both systems suggest the need for further testing based on the probability generated; herein lies their major strength. The application of such systems should help the clinician reach a diagnosis or make a decision as to management in a cost-effective manner.

Bayes Theorem↗

Duration of warfarin anticoagulant therapy and the probabilities of recurrent thromboembolism and hemorrhage.

To determine the association of duration of warfarin anticoagulant therapy after hospitalization for venous thromboembolism with the probability of recurrent thromboembolism and with the risks of hemorrhage, medical records of 2,422 patients hospitalized in 1970 through 1980 with pulmonary embolism, thrombophlebitis, or both were reviewed. Multivariate life-table analyses were performed for 370 patients who had positive results of venography or pulmonary angiography, or who had lung scanning evidence of a "high probability" of pulmonary embolism and no history of the disease. For these patients, warfarin therapy for more than six weeks was not associated with a lower risk of recurrent thromboembolism when compared with warfarin therapy for one through six weeks (risk of recurrence for seven to 26 weeks of treatment 0.8; 95 percent confidence limits 0.3 and 2.5; risk of recurrence for more than 26 weeks of treatment 1.1; 95 percent confidence limits 0.4 and 3.1). The longer the warfarin therapy, the higher the risk of medically important complications from therapy. From one week through five years, the probability of major hemorrhage increased almost linearly: 10 percent for 12 weeks, 18 percent at one year, 26 percent at two years, and 41 percent at five years. This study suggests that intensive, long-term warfarin anticoagulation, in patients with a first episode of venous thromboembolism and no predisposing condition, is associated with more toxicity than efficacy and should be abandoned.

Actuarial Analysis↗

A closer look at the probabilities of the notorious three prisoners.

The "problem of three prisoners", a counterintuitive teaser, is analyzed. It is representative of a class of probability puzzles where the correct solution depends on explication of underlying assumptions. Spontaneous beliefs concerning the problem and intuitive heuristics are reviewed. The psychological background of these beliefs is explored. Several attempts to find a simple criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence are examined. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension of the likelihood-ratio principle (which holds in the case of only two complementary alternatives) to any number of alternatives. Some didactic implications concerning the significance of the chance set-up and reliance on analogies are discussed.

Attention↗

The symptom-association probability: an improved method for symptom analysis of 24-hour esophageal pH data.

BACKGROUND/AIMS: All methods currently used to quantify the temporal relationships between symptoms and episodes of gastroesophageal reflux, as assessed by 24-hour pH monitoring, have major shortcomings. The aim of this study was to develop and validate a simple, all-comprising statistical method to calculate the probability that gastroesophageal reflux episodes and symptoms are associated. METHODS: The 24-hour pH signal was divided into consecutive 2-minute periods. These periods and the 2-minute periods preceding the onset of symptoms were evaluated for the occurrence of reflux. Fisher's Exact Test was then applied to calculate the probability (P value) that reflux and symptom episodes were unrelated. Finally, the symptom-association probability (SAP) was calculated as (1.0 - P) x 100%. The SAP values found in 184 24-hour esophageal pH tests were compared with the symptom index and the symptom sensitivity index. RESULTS: Discordance between the SAP and the symptom index was found in 21 patients (11%) and discordance between the SAP and the symptom-sensitivity index in 28 (15%). False-positive and false-negative symptom index values occurred preferentially in patients with small and large numbers of symptom episodes during the test, respectively (P < 0.05). CONCLUSIONS: The SAP is a single, simple, quantitative measure of the strength of the association between symptoms and reflux episodes that is devoid of the disadvantages inherent to previously used methods.

Adolescent↗

Formula predicting carcinoma-free probability in rats exposed to carcinogen DMBA.

The Weibull distribution is a generalization of the exponential distribution. Pike (Biometrics, 22 (1966) 142-161) used the Weibull distribution in the analysis of carcinogenesis experiments on rats. A formula of conversion that allows mathematical conversion of the Weibull distribution to the generalized lognormal distribution is constructed in this study. This formula is applied to the Pike's data on carcinoma-free probability in rats exposed to carcinogen DMBA. A mathematical model and a computer program previously published by the author are employed in this study. Analysis of the computer-assisted predicted and the Pike's reported data on carcinoma-free probability indicates that a close agreement is present between both data. It seems to the author that a generalized lognormal distribution of the carcinoma-free probability in rats exposed to carcinogen can be derived from the formula of conversion. In addition the generalized lognormal distribution may be proposed as a method of parametric analysis of survival data on estimates by non-parametric product-limit method of Kaplan-Meier in certain biological phenomena.

9,10-Dimethyl-1,2-benzanthracene↗

A new probability mapping method to describe the development of atherosclerotic lesions in cholesterol-fed rabbits.

A new probability mapping method was developed to quantify the size and location of lesions near aortic orifices. The precise location of any part of the lesion could be compared between rabbits. Colour photographs of lesions were projected onto a digitizing tablet, and coded as lesion or non-lesion. Next the orifices were warped onto a standard orifice, and then the lesion mapped to maintain the original length and angular location of the lesion from the edge of the orifice. This method, unlike the previously used polar mapping method, excludes neither absent lesions nor ones which surround more than one orifice. In contrast to other probability mapping methods it warps the orifice rather than the artery wall containing the lesion, and so is easier to use for correlation with histological studies. The eventual aim is to use the probability maps as a tool to estimate the age of various positions of the lesion and to identify areas for histological sampling. The method was used to describe the distribution of lesions in 21 rabbits fed a diet with cholesterol levels declining from 0.5% during the first week, to 0.25% during the next two weeks, to 0.125% for weeks 3-10, to 0.1% for weeks 11-24. This feeding protocol produces fatty lesions which are transformed into fibro-fatty and fibrous lesions with time.

Algorithms↗