Chapter 4. Conditional Probability, Independence, and Bayes′ Theorem

2017 ◽  
pp. 128-155
Keyword(s):  
Conditional Probability ◽  
Bayes Theorem

scholarly journals Probability and Body Composition of Metabolic Syndrome in Young Adults: Use of the Bayes Theorem as Diagnostic Evidence of the Waist-to-Height Ratio

Stats ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Ashuin Kammar ◽  
María Hernández-Hernández ◽  
Patricia López-Moreno ◽  
Angélica Ortíz-Bueno ◽  
María Martínez-Montaño
Keyword(s):  
Metabolic Syndrome ◽  
Young Adults ◽  
Body Composition ◽  
Conditional Probability ◽  
Adult Population ◽  
Bayes Theorem ◽  
Height Ratio ◽  
Anthropometric Parameters ◽  
Anthropometric Parameter ◽  
Waist To Height Ratio

Metabolic syndrome (MS) directly increases the risk of cardiovascular diseases. Childhood and adulthood have been the most studied in MS, leaving aside the young adult population. This study aimed to compare the epidemiological probabilities between MS and different anthropometric parameters of body composition. Using a cross-sectional study with the sample of 1351 young adults, different body composition parameters were obtained such as Waist Circumference (WC), Body Mass Index (BMI), Body Fat% (BF%), Waist-to-Height Ratio (WHtR), and Waist-Hip Ratio. The Bayes Theorem was applied to estimate the conditional probability that any subject developed MS with an altered anthropometric parameter of body composition. Areas under receiver operating characteristic curves (AUCs) and adjusted odds ratios of the five parameters were analyzed in their optimal cutoffs. The conditional probability of developing MS with an altered anthropometric parameter was 17% in WHtR, WC, and Waist-hip R. Furthermore, body composition parameters were adjusted by age, BMI, and gender. Only WHtR (OR = 9.43, CI = 3.4–26.13, p < 0.0001), and BF% (OR = 3.18, CI = 1.42–7.13, p = 0.005) were significant, and the sensitivity (84%) and the AUCs (86%) was higher in WHtR than other parameters. In young adults, the WHtR was the best predictor of metabolic syndrome.

Probability for Everyone—Even Philosophers

2017 ◽  
Author(s):  
Alan Hájek ◽  
Christopher Hitchcock
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Conditional Probability ◽  
Central Limit ◽  
Measure Theory ◽  
Bayes Theorem ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Key Concepts

In this chapter the basics of probability theory are introduced, with particular attention to those topics that are most important for applications in philosophy. The formalism is described in two passes. The first presents finite probability, which suffices for most philosophical discussions of probability. The second presents measure theory, which is needed for applications involving infinities or limits. Key concepts such as conditional probability, probabilistic independence, random variables, and expectation are defined. In addition, several important theorems, including Bayes’ theorem, the weak and strong laws of large numbers, and the central limit theorem are defined. Along the way, several familiar puzzles or paradoxes involving probability are discussed.

On Binomial Observations of Continuous-Time Markovian Population Models

2015 ◽  
Vol 52 (02) ◽  
pp. 457-472 ◽  
Author(s):  
N. G. Bean ◽  
R. Elliott ◽  
A. Eshragh ◽  
J. V. Ross
Keyword(s):  
Conditional Probability ◽  
Continuous Time ◽  
Population Model ◽  
Mass Function ◽  
Bayes Theorem ◽  
Population Models ◽  
Probability Mass Function ◽  
Total Probability ◽  
Birth Process ◽  
Probability Mass

In this paper we consider a class of stochastic processes based on binomial observations of continuous-time, Markovian population models. We derive the conditional probability mass function of the next binomial observation given a set of binomial observations. For this purpose, we first find the conditional probability mass function of the underlying continuous-time Markovian population model, given a set of binomial observations, by exploiting a conditional Bayes' theorem from filtering, and then use the law of total probability to find the former. This result paves the way for further study of the stochastic process introduced by the binomial observations. We utilize our results to show that binomial observations of the simple birth process are non-Markovian.

Using Conditional Probabilities to Understand “Human Error” in Military Aviation Mishaps

2017 ◽  
Vol 61 (1) ◽  
pp. 1619-1620
Author(s):  
Andrew T. Miranda
Keyword(s):  
Conditional Probability ◽  
Department Of Defense ◽  
Classification System ◽  
Human Error ◽  
Bayes Theorem ◽  
Organizational Policy ◽  
Conditional Probabilities ◽  
Technological Environment ◽  
Factors Analysis ◽  
Military Aviation

The Department of Defense Human Factors Analysis and Classification System (DoD HFACS) is the standardized taxonomy used by the services to classify human error identified in mishaps and hazards. DoD HFACS encourages safety investigators and professionals to examine how higher-level influences (e.g. organizational policy, supervisor ability, and technological environment) can impact perfor mance for the individuals executing the actual work (e.g. aviators flying the aircraft). The current project applied the well- known conditional probability formula of Bayes’ Theorem to better understand how the higher-level influences shape the specific types of performance deviations experienced by individuals involved in aviation mishaps.

scholarly journals On Binomial Observations of Continuous-Time Markovian Population Models

2015 ◽  
Vol 52 (2) ◽  
pp. 457-472
Author(s):  
N. G. Bean ◽  
R. Elliott ◽  
A. Eshragh ◽  
J. V. Ross
Keyword(s):  
Conditional Probability ◽  
Continuous Time ◽  
Population Model ◽  
Mass Function ◽  
Bayes Theorem ◽  
Population Models ◽  
Probability Mass Function ◽  
Total Probability ◽  
Birth Process ◽  
Probability Mass

In this paper we consider a class of stochastic processes based on binomial observations of continuous-time, Markovian population models. We derive the conditional probability mass function of the next binomial observation given a set of binomial observations. For this purpose, we first find the conditional probability mass function of the underlying continuous-time Markovian population model, given a set of binomial observations, by exploiting a conditional Bayes' theorem from filtering, and then use the law of total probability to find the former. This result paves the way for further study of the stochastic process introduced by the binomial observations. We utilize our results to show that binomial observations of the simple birth process are non-Markovian.

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