Related papers: The marginalization paradox does not imply inconsi…
Profile likelihood is the key tool for dealing with nuisance parameters in likelihood theory. It is often asserted, however, that profile likelihood is not a 'true' likelihood. One implication is that likelihood theory lacks the generality…
We give a new proof for Godel's second incompleteness theorem, based on Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest…
Consider the following story: A teacher announces to her students a test for the following week, such that the test will be ``surprising''. The students use this as the basis for a ``logical derivation'' and reach a contradiction, which…
How to form priors that do not seem artificial or arbitrary is a central question in Bayesian statistics. The case of forming a prior on the truth of a proposition for which there is no evidence, and the definte evidence that the event can…
We compare two henselisations of a residually discrete valuation domain. Our constructive proof that a certain natural morphism is an isomorphism is also a proof in classical mathematics. Although this isomorphism is implicitly accepted as…
The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable…
It is common in instrumental variable studies for instrument values to be missing, for example when the instrument is a genetic test in Mendelian randomization studies. In this paper we discuss two apparent paradoxes that arise in so-called…
Correlated proportions arise in longitudinal (panel) studies. A typical example is the ``opinion swing'' problem: ``Has the proportion of people favoring a politician changed after his recent speech to the nation on TV?''. Since the same…
Bayesian analyses require that all variable model parameters are given a prior probability distribution. This can pose a challenge for analyses where multiple experiments are combined if these experiments use different parametrisations for…
Hierarchical models in Bayesian inverse problems are characterized by an assumed prior probability distribution for the unknown state and measurement error precision, and hyper-priors for the prior parameters. Combining these probability…
Between the two dominant schools of thought in statistics, namely, Bayesian and classical/frequentist, a main difference is that the former is grounded in the mathematically rigorous theory of probability while the latter is not. In this…
Since Austin introduced the term ``infelicity'', the linguistic literature has been flooded with its use, but no formal or computational explanation has been given for it. This thesis provides one for those infelicities that occur when a…
Recent experiments have perfectly verified the fact that quantum correlations between two entangled particles are stronger than any classical, local pre-quantum worldview allows. This is famously called the EPR paradox first conceived as a…
We investigate the discrimination of two candidates of an unknown parameter in quantum systems with continuous weak measurement, inspired by the application of hypothesis testing in distinguish-ing two Hamiltonians [Kiilerich and M{\o}lmer,…
Standard statistical theory has arguably proved to be unsuitable as a basis for constructing a satisfactory completely general framework for performing statistical inference. For example, frequentist theory has never come close to providing…
A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the…
We show in this note that Gibbs paradox arises not due to application of thermodynamic principles, whether classical or statistical or even quantum mechanical, but due to incorrect application of mathematics to the process of mixing of…
Estimating the difference between two binomial proportions will be investigated, where Bayesian, frequentist and fiducial (BFF) methods will be considered. Three vague priors will be used, the Jeffreys prior, a divergence prior and the…
Optimization is widely used in statistics, and often efficiently delivers point estimates on useful spaces involving structural constraints or combinatorial structure. To quantify uncertainty, Gibbs posterior exponentiates the negative loss…
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…