An Introduction to Bayesian Statistics, Part 1: Conditional Probabilities and Bayes Theorem

NIR news ◽  
2012 ◽  
Vol 23 (3) ◽  
pp. 18-19 ◽  
Author(s):  
Tom Fearn
Keyword(s):  
Bayesian Statistics ◽  
Bayes Theorem ◽  
Conditional Probabilities

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 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.

The Once-ler Problem: Introduction to Decision Trees

2019 ◽  
pp. 353-368
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Decision Tree ◽  
Bayesian Networks ◽  
Decision Trees ◽  
Expected Utility ◽  
Decision Problem ◽  
Graphical Representation ◽  
Bayes Theorem ◽  
Conditional Probabilities ◽  
Expected Values ◽  
The Lorax

In the “Once-ler Problem,” the decision tree is introduced as a very useful technique that can be used to answer a variety of questions and assist in making decisions. This chapter builds on the “Lorax Problem” introduced in Chapter 19, where Bayesian networks were introduced. A decision tree is a graphical representation of the alternatives in a decision. It is closely related to Bayesian networks except that the decision problem takes the shape of a tree instead. The tree itself consists of decision nodes, chance nodes, and end nodes, which provide an outcome. In a decision tree, probabilities associated with chance nodes are conditional probabilities, which Bayes’ Theorem can be used to estimate or update. The calculation of expected values (or expected utility) of competing alternative decisions is provided on a step-by-step basis with an example from The Lorax.

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.

Probability and random variables

2019 ◽  
pp. 38-59
Author(s):  
M. D. Edge
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Bayes Theorem ◽  
Probability Density Functions ◽  
Density Functions ◽  
Conditional Probabilities ◽  
Discrete Random Variables ◽  
New Information ◽  
Probability Mass ◽  
Mass Functions

This chapter considers the rules of probability. Probabilities are non-negative, they sum to one, and the probability that either of two mutually exclusive events occurs is the sum of the probability of the two events. Two events are said to be independent if the probability that they both occur is the product of the probabilities that each event occurs. Bayes’ theorem is used to update probabilities on the basis of new information, and it is shown that the conditional probabilities P(A|B) and P(B|A) are not the same. Finally, the chapter discusses ways in which distributions of random variables can be described, using probability mass functions for discrete random variables and probability density functions for continuous random variables.

Bayesian Statistics

2021 ◽  
pp. 165-180
Author(s):  
Timothy E. Essington
Keyword(s):  
Parameter Estimation ◽  
Statistical Analysis ◽  
Model Selection ◽  
Bayesian Statistics ◽  
Bayes Theorem ◽  
Bayesian Parameter Estimation ◽  
Bayesian Approaches ◽  
Frequentist Statistics ◽  
The Difference ◽  
The Bayesian Approach

The chapter “Bayesian Statistics” gives a brief overview of the Bayesian approach to statistical analysis. It starts off by examining the difference between frequentist statistics and Bayesian statistics. Next, it introduces Bayes’ theorem and explains how the theorem is used in statistics and model selection, with the prosecutor’s fallacy given as a practice example. The chapter then goes on to discuss priors and Bayesian parameter estimation. It concludes with some final thoughts on Bayesian approaches. The chapter does not answer the question “Should ecologists become Bayesian?” However, to the extent that alternative models can be posed as alternative values of parameters, Bayesian parameter estimation can help assign probabilities to those hypotheses.

scholarly journals Electron Energy Distributions of AGNs in the Thin Synchrotron Limit. I. the Method

1994 ◽  
Vol 159 ◽  
pp. 358-359
Author(s):  
Luis Salas ◽  
Irene Cruz-González ◽  
Luis Carrasco
Keyword(s):  
Synchrotron Radiation ◽  
Energy Distribution ◽  
Electron Energy ◽  
Bayes Theorem ◽  
Spectral Energy ◽  
Conditional Probabilities ◽  
Spectral Energy Distributions ◽  
Energy Distributions ◽  
Self Consistent ◽  
Consistent Solution

We develop a new method called “Inverse Synchrotron Transform” (IST) to study the spectral energy distributions of AGNs. We demonstrate that it is possible to use Bayes Theorem for conditional probabilities to derive a self-consistent solution for the electron energy distribution (EEDs), starting from the observed spectral energy distributions (SEDs) and the assumption that the only physical process involved is thin synchrotron radiation. We test the IST method and find that it allows to distinguish among different EEDs that produce SEDs which nevertheless seem very similar. We apply the method to multifrequency simultaneous observations of AGNs (paper II, this conference).

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.

Bayesian Basics and the Scientific Hypothesis

2019 ◽  
pp. 143-160
Author(s):  
Bradley E. Alger
Keyword(s):  
Model Selection ◽  
Hypothesis Testing ◽  
Conceptual Framework ◽  
Bayesian Statistics ◽  
Bayesian Methods ◽  
Bayes Theorem ◽  
Testing Framework ◽  
Parameter Evaluation ◽  
Bayesian Procedures ◽  
Scientific Hypothesis

This chapter covers the basics of Bayesian statistics, emphasizing the conceptual framework for Bayes’ Theorem. It works through several iterations of the theorem to demonstrate how the same equation is applied in different circumstances, from constructing and updating models to parameter evaluation, to try to establish an intuitive feel for it. The chapter also covers the philosophical underpinnings of Bayesianism and compares them with the frequentist perspective described in Chapter 5. It addresses the question of whether Bayesians are inductivists. Finally, the chapter shows how the Bayesian procedures of model selection and comparison can be pressed into service to allow Bayesian methods to be used in hypothesis testing in essentially the same way that various p-tests are used in the frequentist hypothesis testing framework.

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