A Type Theory for Probabilistic and Bayesian Reasoning

This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference

Uncertainty reasoning and representation: A Comparison of several alternative approaches

Much of the research done in Artificial Intelligence involves investigating and developing methods of incorporating uncertainty reasoning and representation into expert systems. Several methods have been proposed and attempted for handling uncertainty in problem solving situations. The theories range from numerical approaches based on strict probabilistic reasoning to non-numeric approaches based on logical reasoning. This study investigates a number of these approaches including Bayesian Probability, Mycin Certainty Factors, Dempster-Shafer Theory of Evidence, Fuzzy Set Theory, Possibility Theory and non monotonic logic. Each of these theories and their underlying formalisms are explored by means of examples. The discussion concentrates on a comparison of the different approaches, noting the type of uncertainty that they best represent

A probabilistic model for information and sensor validation

This paper develops a new theory and model for information and sensor validation. The model represents relationships between variables using Bayesian networks and utilizes probabilistic propagation to estimate the expected values of variables. If the estimated value of a variable differs from the actual value, an apparent fault is detected. The fault is only apparent since it may be that the estimated value is itself based on faulty data. The theory extends our understanding of when it is possible to isolate real faults from potential faults and supports the development of an algorithm that is capable of isolating real faults without deferring the problem to the use of expert provided domain-specific rules. To enable practical adoption for real-time processes, an any time version of the algorithm is developed, that, unlike most other algorithms, is capable of returning improving assessments of the validity of the sensors as it accumulates more evidence with time. The developed model is tested by applying it to the validation of temperature sensors during the start-up phase of a gas turbine when conditions are not stable; a problem that is known to be challenging. The paper concludes with a discussion of the practical applicability and scalability of the model

The Bayesian sampler : generic Bayesian inference causes incoherence in human probability

Human probability judgments are systematically biased, in apparent tension with Bayesian models of cognition. But perhaps the brain does not represent probabilities explicitly, but approximates probabilistic calculations through a process of sampling, as used in computational probabilistic models in statistics. Naïve probability estimates can be obtained by calculating the relative frequency of an event within a sample, but these estimates tend to be extreme when the sample size is small. We propose instead that people use a generic prior to improve the accuracy of their probability estimates based on samples, and we call this model the Bayesian sampler. The Bayesian sampler trades off the coherence of probabilistic judgments for improved accuracy, and provides a single framework for explaining phenomena associated with diverse biases and heuristics such as conservatism and the conjunction fallacy. The approach turns out to provide a rational reinterpretation of “noise” in an important recent model of probability judgment, the probability theory plus noise model (Costello & Watts, 2014, 2016a, 2017; Costello & Watts, 2019; Costello, Watts, & Fisher, 2018), making equivalent average predictions for simple events, conjunctions, and disjunctions. The Bayesian sampler does, however, make distinct predictions for conditional probabilities and distributions of probability estimates. We show in 2 new experiments that this model better captures these mean judgments both qualitatively and quantitatively; which model best fits individual distributions of responses depends on the assumed size of the cognitive sample

Probabilistic biases meet the Bayesian brain

Bayesian cognitive science sees the mind as a spectacular probabilistic inference machine. But Judgment and Decision Making research has spent half a century uncovering how dramatically and systematically people depart from rational norms. This paper outlines recent research that opens up the possibility of an unexpected reconciliation. The key hypothesis is that the brain neither represents nor calculates with probabilities; but approximates probabilistic calculations through drawing samples from memory or mental simulation. Sampling models diverge from perfect probabilistic calculations in ways that capture many classic JDM findings, and offers the hope of an integrated explanation of classic heuristics and biases, including availability, representativeness, and anchoring and adjustment