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We marshall the arguments for preferring Bayesian hypothesis testing and confidence sets to frequentist ones. We define admissible solutions to inference problems, noting that Bayesian solutions are admissible. We give seven weaker…
We study conditioning on null events, or surprises, and behaviorally characterize the Ordered Surprises (OS) representation of beliefs. For feasible events, our Decision Maker (DM) is Bayesian. For null events, our DM considers a hierarchy…
The method of maximum entropy has proven to be a rather powerful way to solve the inverse problem consisting of determining a probability density $f_S(s)$ on $[0,\infty)$ from the knowledge of the expected value of a few generalized…
Probability models are only useful at explaining the uncertainty of what we do not know, and should never be used to say what we already know. Probability and statistical models are useless at discerning cause. Classical statistical…
Statistical learning using imprecise probabilities is gaining more attention because it presents an alternative strategy for reducing irreplicable findings by freeing the user from the task of making up unwarranted high-resolution…
The arrival of AI techniques in computations, with the potential for hallucinations and non-robustness, has made trustworthiness of algorithms a focal point. However, trustworthiness of the many classical approaches are not well understood.…
We revisit the problem of general identifiability originally introduced in [Lee et al., 2019] for causal inference and note that it is necessary to add positivity assumption of observational distribution to the original definition of the…
The reversibility and recurrence paradoxes are key issues that have been left unsolved in researches on the foundation of thermodynamics since the 19th century. This article shows that (1) the reversibility paradox can be overcome if we pay…
We present a propositional logic %which can be used to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and…
An agent observes a clue, and an analyst observes an inference: a ranking of events on the basis of how corroborated they are by the clue. We prove that if the inference satisfies the axioms of Villegas (1964) except for the classic…
Proof by induction plays a central role in formal verification. However, its automation remains as a formidable challenge in Computer Science. To solve inductive problems, human engineers often have to provide auxiliary lemmas manually. We…
Contemporary approaches frame the color constancy problem as learning camera specific illuminant mappings. While high accuracy can be achieved on camera specific data, these models depend on camera spectral sensitivity and typically exhibit…
Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum…
Richard Cox [1] set the axiomatic foundations of probable inference and the algebra of propositions. He showed that consistency within these axioms requires certain rules for updating belief. In this paper we use the analogy between…
Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative…
Conditional selective inference requires an exact characterization of the selection event, which is often unavailable except for a few examples like the lasso. This work addresses this challenge by introducing a generic approach to estimate…
The analysis of theory-confirmation generally takes the deductive form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure…
Machine learning models have undeniably achieved impressive performance across a range of applications. However, their often perceived black-box nature, and lack of transparency in decision-making, have raised concerns about understanding…
Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive…
Determining the initial state of the universe is a challenging problem in quantum cosmology and we argue that the issue is intractable if the basic postulates of quantum mechanics are not modified in a nontrivial way. Namely a "standard"…