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The main object of this paper is to present some general concepts of Bayesian inference and more specifically the estimation of the hyperparameters in inverse problems. We consider a general linear situation where we are given some data…
A generalized Kullback-Leibler relative entropy is introduced starting with the symmetric Jackson derivative of the generalized overlap between two probability distributions. The generalization retains much of the structure possessed by the…
The density matrix yields probabilistic information about the outcome of measurements on a quantum system, but it does not distinguish between classical randomness in the preparation of the system and entanglement with its environment.…
We discuss the relation between density matrices and the uncertainty principle; this allows us to justify and explain a recent statement by Man'ko et al. We thereafter use Hardy's uncertainty principle to prove a new result for Wigner…
Concerning systematic effects, the recommendation given in the GUM is to correct for them, but unfortunately no detailed information is available, how to do this. This publication will show, how systematic measurement deviations can be…
This paper explores Bayesian estimation for categorical data, focusing on simple yet effective models that provide a foundation for applying more advanced methods accurately and reliably in real-world applications. We begin by revisiting…
Generalized linear models (GLMs) are routinely used for modeling relationships between a response variable and a set of covariates. The simple form of a GLM comes with easy interpretability, but also leads to concerns about model…
Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how fundamental concepts…
We propose a novel method for closed-form predictive distribution modeling with neural nets. In quantifying prediction uncertainty, we build on Evidential Deep Learning, which has been impactful as being both simple to implement and giving…
Bayesian sequence prediction is a simple technique for predicting future symbols sampled from an unknown measure on infinite sequences over a countable alphabet. While strong bounds on the expected cumulative error are known, there are only…
Tree-based priors for probability distributions are usually specified using a predetermined, data-independent collection of candidate recursive partitions of the sample space. To characterize an unknown target density in detail over the…
Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term "likelihood-free" refers to problems where the likelihood is intractable to compute or estimate directly, but where it…
We show that any quantum density matrix can be represented by a Bayesian network (a directed acyclic graph), and also by a Markov network (an undirected graph). We show that any Bayesian or Markov net that represents a density matrix, is…
In the simple vs composite hypothesis test with a proper prior, the Bayes Factor (BF) is shown to be the posterior mean of the Likelihood Ratio (LR). Therefore, the posterior standard deviation of the LR or rather its posterior cumulative…
The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to…
Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions…
We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often…
How might a smooth probability distribution be estimated, with accurately quantified uncertainty, from a limited amount of sampled data? Here we describe a field-theoretic approach that addresses this problem remarkably well in one…
We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One…
We introduce a Bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights.…