scholarly journals Characterization of the Bayesian Posterior Distribution in Terms of Self-information

2017 ◽  
Vol 7 (1) ◽  
pp. 21
Author(s):  
Marco Dall'Aglio ◽  
Theodore P. Hill
Keyword(s):  
Bayesian Statistics ◽  
Unique Solution ◽  
Posterior Distribution ◽  
Optimization Problems ◽  
Probability Distributions ◽  
Bayes Theorem ◽  
Conditional Probabilities ◽  
Loss Of Self

It is well known that the classical Bayesian posterior arises naturally as the unique solution of different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. Here it is shown that the Bayesian posterior is also the unique minimax optimizer of the loss of self-information in combining the prior and the likelihood distributions, and is the unique proportional consolidation of the same distributions. These results, direct corollaries of recent results about conflations of probability distributions, further reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.

Overview of Bayesian Statistics

2020 ◽  
pp. 0193841X1989562
Author(s):  
David Rindskopf
Keyword(s):  
Bayesian Inference ◽  
Bayesian Statistics ◽  
Statistical Modeling ◽  
Probability Distributions ◽  
Bayes Theorem ◽  
Special Issue ◽  
Popular Approach ◽  
New Information ◽  
Basic Concepts ◽  
Phd Program

Bayesian statistics is becoming a popular approach to handling complex statistical modeling. This special issue of Evaluation Review features several Bayesian contributions. In this overview, I present the basics of Bayesian inference. Bayesian statistics is based on the principle that parameters have a distribution of beliefs about them that behave exactly like probability distributions. We can use Bayes’ Theorem to update our beliefs about values of the parameters as new information becomes available. Even better, we can make statements that frequentists do not, such as “the probability that an effect is larger than 0 is .93,” and can interpret 95% (e.g.) intervals as people naturally want, that there is a 95% probability that the parameter is in that interval. I illustrate the basic concepts of Bayesian statistics through a simple example of predicting admissions to a PhD program.

Glossary of terms

2010 ◽  
pp. 500-506
Author(s):  
Janet L. Peacock ◽  
Philip J. Peacock
Keyword(s):  
Bayesian Statistics ◽  
Analysis Of Variance ◽  
Prior Information ◽  
Statistical Approach ◽  
Bayes Theorem ◽  
Conditional Probabilities ◽  
Unknown Parameters ◽  
Bayes’S Theorem ◽  
Bayes's Theorem

Analysis of variance See One-way analysis of variance (p. 280) and Two-way analysis of variance (p. 412) Bayes’s theorem A formula that allows the reversal of conditional probabilities (see Bayes’ theorem, p. 234) Bayesian statistics A statistical approach based on Bayes’ theorem, where prior information or beliefs are combined with new data to provide estimates of unknown parameters (see ...

scholarly journals Identical twins and Bayes' theorem in the 21st century

F1000Research ◽  
2013 ◽  
Vol 2 ◽  
pp. 278
Author(s):  
Valentin Amrhein ◽  
Tobias Roth ◽  
Fränzi Korner-Nievergelt
Keyword(s):  
Bayesian Statistics ◽  
21St Century ◽  
Posterior Distribution ◽  
Recent Article ◽  
Bayes Theorem ◽  
Identical Twins ◽  
Data Point

In a recent article in Science on "Bayes' Theorem in the 21st Century", Bradley Efron uses Bayes' theorem to calculate the probability that twins are identical given that the sonogram shows twin boys. He concludes that Bayesian calculations cannot be uncritically accepted when using uninformative priors. We argue that this conclusion is problematic because Efron's example on identical twins does not use data, hence it is not Bayesian statistics; his priors are not appropriate and are not uninformative; and using the available data point and an uninformative prior actually leads to a reasonable posterior distribution.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

scholarly journals Characterization of Degenerate Configurations in Attitude Determination of Three-Vehicle Heterogeneous Formations

Sensors ◽  
2021 ◽  
Vol 21 (14) ◽  
pp. 4631
Author(s):  
Pedro Cruz ◽  
Pedro Batista
Keyword(s):  
Unique Solution ◽  
Multiple Solutions ◽  
Attitude Determination ◽  
Line Of Sight ◽  
Number Of Solutions ◽  
Relative Line ◽  
Heterogeneous Formations ◽  
Heterogeneous Formation

The existence of multiple solutions to an attitude determination problem impacts the design of estimation schemes, potentially increasing the errors by a significant value. It is therefore essential to identify such cases in any attitude problem. In this paper, the cases where multiple attitudes satisfy all constraints of a three-vehicle heterogeneous formation are identified. In the formation considered herein, the vehicles measure inertial references and relative line-of-sight vectors. Nonetheless, the line of sight between two elements of the formation is restricted, and these elements are denoted as deputies. The attitude determination problem is characterized relative to the number of solutions associated with each configuration of the formation. There are degenerate and ambiguous configurations that result in infinite or exactly two solutions, respectively. Otherwise, the problem has a unique solution. The degenerate configurations require some collinearity between independent measurements, whereas the ambiguous configurations result from symmetries in the formation measurements. The conditions which define all such configurations are determined in this work. Furthermore, the ambiguous subset of configurations is geometrically interpreted resorting to the planes defined by specific measurements. This subset is also shown to be a zero-measure subset of all possible configurations. Finally, a maneuver is simulated to illustrate and validate the conclusions. As a result of this analysis, it is concluded that, in general, the problem has one attitude solution. Nonetheless, there are configurations with two or infinite solutions, which are identified in this work.

scholarly journals A Brief Review of Generalized Entropies

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 813 ◽  
Author(s):  
José Amigó ◽  
Sámuel Balogh ◽  
Sergio Hernández
Keyword(s):  
Diffusion Processes ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Point Of View ◽  
Scaling Exponent ◽  
Scaling Exponents ◽  
Physical Systems ◽  
New Phenomena ◽  
Generalized Entropies

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

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