Skip to content

Explore Courses | Elder Research | Contact | LMS Login

Statistics.com Logo
  • Courses
    • See All Courses
    • Calendar
    • Intro stats for college credit
    • Faculty
    • Group training
    • Credit & Credentialing
    • Teach With Us
  • Programs/Degrees
    • Certificates
      • Analytics for Data Science
      • Biostatistics
      • Programming For Data Science – Python (Experienced)
      • Programming For Data Science – Python (Novice)
      • Programming For Data Science – R (Experienced)
      • Programming For Data Science – R (Novice)
      • Social Science
    • Undergraduate Degree Programs
    • Graduate Degree Programs
    • Massive Open Online Courses (MOOC)
  • Partnerships
    • Higher Education
    • Enterprise
  • Resources
    • About Us
    • Blog
    • Word Of The Week
    • News and Announcements
    • Newsletter signup
    • Glossary
    • Statistical Symbols
    • FAQs & Knowledge Base
    • Testimonials
    • Test Yourself
Menu
  • Courses
    • See All Courses
    • Calendar
    • Intro stats for college credit
    • Faculty
    • Group training
    • Credit & Credentialing
    • Teach With Us
  • Programs/Degrees
    • Certificates
      • Analytics for Data Science
      • Biostatistics
      • Programming For Data Science – Python (Experienced)
      • Programming For Data Science – Python (Novice)
      • Programming For Data Science – R (Experienced)
      • Programming For Data Science – R (Novice)
      • Social Science
    • Undergraduate Degree Programs
    • Graduate Degree Programs
    • Massive Open Online Courses (MOOC)
  • Partnerships
    • Higher Education
    • Enterprise
  • Resources
    • About Us
    • Blog
    • Word Of The Week
    • News and Announcements
    • Newsletter signup
    • Glossary
    • Statistical Symbols
    • FAQs & Knowledge Base
    • Testimonials
    • Test Yourself
Student Login

Prior and posterior probability (difference)

Prior and posterior probability (difference)

Prior and posterior probability (difference):

Consider a population where the proportion of HIV-infected individuals is 0.01. Then, the prior probability that a randomly chosen subject is HIV-infected is Pprior = 0.01 .


Suppose now a subject has been positive for HIV. It is known that specificity of the test is 95%, and sensitivity of the test is 99%. What is the probability that the subject is HIV-infected? In other words, what is the conditional probability that a subject is HIV-infected if he/she has tested positive?


The following table summarizes calculations. For the sake of simplicity you may consider the fractions (probabilities) as proportions of the general population.


. Test results .
HIV-status Positive Negative Total
infected 0.01*0.99 = 0.0099 0.01*(1-0.99)=0.0001 0.01
not infected 0.99*(1-0.95) 0.99*(0.95) 0.99
total: 0.0594 0.9406 1.00


Thus, the average proportion of positive tests overall is 0.0594, and the proportion of actually infected among them is 0.0099/0.0594 or 0.167 = 16.7%.


So, the posterior (i.e. after the test has been carried out and turns out to be positive) probability that the subject is really HIV-infected is 0.167.


The difference between prior and posterior probabilities characterizes the information we have gotten from the experiment or measurement. In this example the probability changed from 0.01 (prior) to 0.167 (posterior)


Note also the surprising result in this case, which, although hypothetical, is typical of many medical screening tests. Although the test is 95% effective in correctly identifying an HIV case, a person who tests positive actually has only a 16.7% chance of having the disease. This is due to the very low proportion of actually-infected people in the population -- most of the positive test results are false positives from the non-infected people who are being tested.


See also: A Priori Probability , Posterior Probability .

Browse Other Glossary Entries

Courses Using This Term

Loading...
Introduction to Bayesian Statistics
This course will teach you the basic ideas of Bayesian Statistics: how to perform Bayesian analysis for a binomial proportion, a normal mean, the difference between normal means, the difference between proportions, and for a simple linear regression model.
Return to Glossary Search

About Statistics.com

Statistics.com offers academic and professional education in statistics, analytics, and data science at beginner, intermediate, and advanced levels of instruction. Statistics.com is a part of Elder Research, a data science consultancy with 25 years of experience in data analytics.

 The Institute for Statistics Education is certified to operate by the State Council of Higher Education for Virginia (SCHEV)

Our Links

  • Contact Us
  • Site Map
  • Explore Courses
  • About Us
  • Management Team
  • Contact Us
  • Site Map
  • Explore Courses
  • About Us
  • Management Team

Social Networks

Facebook Twitter Youtube Linkedin

Contact

The Institute for Statistics Education
2107 Wilson Blvd
Suite 850 
Arlington, VA 22201
(571) 281-8817

ourcourses@statistics.com

  • Contact Us
  • Site Map
  • Explore Courses
  • About Us
  • Management Team

© Copyright 2023 - Statistics.com, LLC | All Rights Reserved | Privacy Policy | Terms of Use

By continuing to use this website, you consent to the use of cookies in accordance with our Cookie Policy.

Accept