Question: Describe The Concepts Of Prior Probability And Con

Question: Describe the concepts of prior probability, conditional probability, joint probability, and posterior probability. Explain Bayes’ Theorem and provide a numerical example related to a real-life situation. Note: 1. Need to write at least 2 paragraphs 2. Need to include the information from the textbook as the reference. 3. Need to include at least 1 peer reviewed article as the reference. 4. Please find the textbook and related power point in the attachment

Describe The Concepts Of Prior Probability Conditional Proba

Question: Describe the concepts of prior probability, conditional probability, joint probability, and posterior probability. Explain Bayes’ Theorem and provide a numerical example related to a real-life situation. Note: 1. Need to write at least 2 paragraphs 2. Need to include the information from the textbook as the reference. 3. Need to include at least 1 peer reviewed article as the reference. 4. Please find the textbook and related power point in the attachment

Paper For Above instruction

Bayes' Theorem and the related probabilistic concepts—prior, conditional, joint, and posterior probabilities—form a foundational framework in statistical inference, especially pertinent in fields such as medicine, finance, and machine learning. Understanding these concepts begins with prior probability, which represents the initial assessment of the likelihood of an event before considering new evidence. For example, the prior probability could be the estimated prevalence of a disease in a population. Conditional probability, on the other hand, describes the probability of an event occurring given that another event has already occurred, such as the probability of testing positive for a disease given that a person actually has the disease. Joint probability measures the likelihood of two events happening simultaneously and is essential in understanding the combined occurrence of two variables. Posterior probability then updates the prior belief after considering new evidence, effectively refining the initial estimate based on observed data using Bayes' Theorem.

Bayes' Theorem provides a mathematical framework for updating the probability estimate of an event based on new information. It is expressed as: P(A|B) = [P(B|A) * P(A)] / P(B), where P(A|B) is the posterior probability, P(B|A) is the likelihood of evidence given the hypothesis, P(A) is the prior probability, and P(B) is the total probability of the evidence. A practical illustration of Bayes' Theorem can be seen in medical diagnostics: suppose a test for a disease has a 99% sensitivity and 95% specificity. If the disease prevalence (prior) in the population is 1%, applying Bayes' Theorem helps determine the actual probability that a person testing positive truly has the disease, which is often lower than the test’s accuracy alone due to the low prevalence. This example underscores the importance of Bayesian reasoning in making informed decisions based on probabilistic evidence, especially when dealing with rare conditions.

References

  • Feller, W. (1957). An Introduction to Probability Theory and Its Applications. Wiley.
  • Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
  • McGrayne, S. B. (2011). The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Reinvented in the Digital Age. Yale University Press.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.
  • Kruschke, J. K. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan. Academic Press.
  • Provost, F., & Fawcett, T. (2013). Data Science for Business: What You Need to Know about Data Mining and Data-Analytic Thinking. O'Reilly Media.
  • Lindley, D. V. (2000). Understanding Uncertainty. Wiley.
  • Robert, C. P. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer.
  • Albert, J. H. (2008). Bayesian Computation with R. Springer.
  • Albert, J. H., & Chib, S. (1993). Bayesian Analysis of Binary and Polychotomous Response Data. Journal of the American Statistical Association, 88(422), 669–679.

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