Frontiers in Psychology / Imprecise Uncertain Reasoning : A Distributional Approach

The contribution proposes to model imprecise and uncertain reasoning by a mental probability logic that is based on probability distributions. It shows how distributions are combined with logical operators and how distributions propagate in inference rules. It discusses a series of examples like the Linda task, the suppression task, Doherty's pseudodiagnosticity task, and some of the deductive reasoning tasks of Rips. It demonstrates how to update distributions by soft evidence and how to represent correlated risks. The probabilities inferred from different logical inference forms may be so similar that it will be impossible to distinguish them empirically in a psychological study. Second-order distributions allow to obtain the probability distribution of being coherent. The maximum probability of being coherent is a second-order criterion of rationality. Technically the contribution relies on beta distributions, copulas, vines, and stochastic simulation.(VLID)311645

Modeling biased information seeking with second order probability distributions

summary:Updating probabilities by information from only one hypothesis and thereby ignoring alternative hypotheses, is not only biased but leads to progressively imprecise conclusions. In psychology this phenomenon was studied in experiments with the “pseudodiagnosticity task”. In probability logic the phenomenon that additional premises increase the imprecision of a conclusion is known as “degradation”. The present contribution investigates degradation in the context of second order probability distributions. It uses beta distributions as marginals and copulae together with C-vines to represent dependence structures. It demonstrates that in Bayes' theorem the posterior distributions of the lower and upper probabilities approach 0 and 1 as more and more likelihoods belonging to only one hypothesis are included in the analysis

Propagating Imprecise Probabilities In Bayesian Networks

Often experts are incapable of providing `exact' probabilities; likewise, samples on which the probabilities in networks are based must often be small and preliminary. In such cases the probabilities in the networks are imprecise. The imprecision can be handled by second-order probability distributions. It is convenient to use beta or Dirichlet distributions to express the uncertainty about probabilities. The problem of how to propagate point probabilities in a Bayesian network now is transformed into the problem of how to propagate Dirichlet distributions in Bayesian networks. It is shown that the propagation of Dirichlet distributions in Bayesian networks with incomplete data results in a system of probability mixtures of beta-binomial and Dirichlet distributions. Approximate first order probabilities and their second order probability density functions are be obtained by stochastic simulation. A number of properties of the propagation of imprecise probabilities are discuss..

Perfect Sequences: A Contribution to Structuring Conditional Independence Models

The representation of conditional independence models by perfect sequences provides an alternative to Bayesian networks and essential graphs. The paper discusses properties of perfect sequences that are relevant with respect to different structures of conditional independence models. Boundary variables (related to terminal nodes in a directed graph representation) are used to find the number of labeled and unlabeled models and to enumerate parts of the model space. Structuring principles are further applied to the evaluation of whole conditional independence models in learning models from data.

Exchangeability in Probability Logic

The paper investigates exchangeability in the context of probability logic. We study generalizations of basic inference rules and inferences involving cardinalities. We compare the results with those obtained in the case in which only identical probabilities are assumed

Psychologica Belgica / Towards a mental probability logic

We propose probability logic as an appropriate standard of reference for evaluating human inferences. Probability logical accounts of nonmonotonic reasoning with SYSTEM P, and conditional syllogisms (MODUS PONENS, etc.) are explored. Furthermore, we present categorical syllogisms with intermediate quantifiers, like the "MOST " quantifier. While most of the paper is theoretical and intended to stimulate psychological studies, we also summarize our empirical studies on human nonmonotonic reasoning.(VLID)221355