Related papers: Probability Theory without Bayes' Rule
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…
When testing a statistical hypothesis, is it legitimate to deliberate on the basis of initial data about whether and how to collect further data? Game-theoretic probability's fundamental principle for testing by betting says yes, provided…
The ``impossibility theorem'' -- which is considered foundational in algorithmic fairness literature -- asserts that there must be trade-offs between common notions of fairness and performance when fitting statistical models, except in two…
In inference problems involving a multi-dimensional parameter $\theta$, it is often natural to consider decision rules that have a risk which is invariant under some group $G$ of permutations of $\theta$. We show that this implies that the…
A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of…
We present a comparative study between classical probability and quantum probability from the Bayesian viewpoint, where probability is construed as our rational degree of belief on whether a given statement is true. From this viewpoint,…
Inference is the process of using facts we know to learn about facts we do not know. A theory of inference gives assumptions necessary to get from the former to the latter, along with a definition for and summary of the resulting…
Given a universe of discourse X-a domain of possible outcomes-an experiment may consist of selecting one of its elements, subject to the operation of chance, or of observing the elements, subject to imprecision. A priori uncertainty about…
There are three principle paradigms of statistical inference: (i) Bayesian, (ii) information-based and (iii) frequentist inference. We describe an objective prior (the weighting or $w$-prior) which unifies objective Bayes and…
After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference…
We develop a framework for the operationalization of models and parameters by combining de Finetti's representation theorem with a conditional form of Sanov's theorem. This synthesis, the tilted de Finetti theorem, shows that conditioning…
It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such…
The definition of the conditional probability is very important in the theory of the probability. This definition is based on the fact, that random events can be simultaneously measurable. This paper deal with the problem of conditioning…
Bayesian hypothesis testing is re-examined from the perspective of an a priori assessment of the test statistic distribution under the alternative. By assessing the distribution of an observable test statistic, rather than prior parameter…
When dealing with Bayesian inference the choice of the prior often remains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the…
It is suggested that an AI inference system should reflect an inference policy that is tailored to the domain of problems to which it is applied -- and furthermore that an inference policy need not conform to any general theory of rational…
The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is…
We consider the problem of inferring the probability distribution associated with a language, given data consisting of an infinite sequence of elements of the languge. We do this under two assumptions on the algorithms concerned: (i) like a…
This paper gives new foundations of quantum state reduction without appealing to the projection postulate for the probe measurement. For this purpose, the quantum Bayes principle is formulated as the most fundamental principle for…
In the deterministic context Bakushinskii's theorem excludes the existence of purely data driven convergent regularization for ill-posed problems. We will prove in the present work that in the statistical setting we can either construct a…