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We consider a simple model of imprecise comparisons: there exists some $\delta>0$ such that when a subject is given two elements to compare, if the values of those elements (as perceived by the subject) differ by at least $\delta$, then the…
Human inertial thinking schemes can be formed through learning, which are then applied to quickly solve similar problems later. However, when problems are significantly different, inertial thinking generally presents the solutions that are…
Inverse classification, the process of making meaningful perturbations to a test point such that it is more likely to have a desired classification, has previously been addressed using data from a single static point in time. Such an…
This paper proposes a careful separation between an entity's epistemic system and their decision system. Crucially, Bayesian counterfactuals are estimated by the epistemic system; not by the decision system. Based on this remark, I prove…
The purpose of this paper is to provide a random version of Simons' inequality.
I give a simple analysis of the game that I previously published in Scientific American which shows the paradoxical behavior whereby two losing games randomly combine to form a winning game. The game, modeled on a random walk, requires only…
Defeasible reasoning is the mode of reasoning where conclusions can be overturned by taking into account new evidence. A commonly used method in cognitive science and logic literature is to handcraft argumentation supporting inference…
"The chance to win given a certain move" is an easily obtainable quantity from data and often quoted in gaming statistics. It is also the fundamental quantity that reinforcement learning AI bases on. Unfortunately, this conditional…
We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player andyet the new combined game is favourable -- this is a counter-intuitive…
The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
The goal of a scientific investigation is to find answers to specific questions. In geosciences this is typically achieved by solving an inference or inverse problem and interpreting the solution. However, the answer obtained is often…
The derivation of the quantum retrodictive probability formula involves an error, an ambiguity. The end result is correct because this error appears twice, in such a way as to cancel itself. In addition, however, the usual expression for…
Comparing the differences in outcomes (that is, in "dependent variables") between two subpopulations is often most informative when comparing outcomes only for individuals from the subpopulations who are similar according to "independent…
The discrepant posterior phenomenon (DPP) is a counter-intuitive phenomenon that can frequently occur in a Bayesian analysis of multivariate parameters. It refers to the phenomenon that a parameter estimate based on a posterior is more…
Counterfactual explanations are usually obtained by identifying the smallest change made to an input to change a prediction made by a fixed model (hereafter called sparse methods). Recent work, however, has revitalized an old insight: there…
We prove an inequality on positive real numbers, that looks like a reverse to the well-known Hilbert inequality, and we use some unusual techniques from Fourier analysis to prove that this inequality is optimal.
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…