Bayesian or not Bayesian,this is my problem

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Discussion Overview

The discussion revolves around the application of Bayesian reasoning in quality assurance for a set of elements, particularly focusing on calculating posterior probabilities based on test results that include false positives and false negatives. The scope includes theoretical aspects of Bayesian statistics and practical implications for prioritizing elements based on their estimated probabilities of correctness.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a scenario involving a test with known false positive and false negative rates, seeking to calculate the posterior probability of an element being correct given a test estimate of 75% correctness.
  • The same participant expresses uncertainty about applying Bayesian theorem due to a lack of knowledge regarding the number of correct elements, proposing an instinctual frequency-based approach instead.
  • Another participant questions the clarity and completeness of the initial question, asking if it is derived from a specific source or if data is missing.
  • A third participant suggests a method for checking calculations by establishing a relationship between defective parts and test scores, presenting a formulaic approach to derive conditional probabilities.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the initial question's clarity or the correctness of the proposed calculations. Multiple viewpoints and methods are presented without resolution.

Contextual Notes

There is uncertainty regarding the completeness of the data provided in the initial question, as well as the assumptions underlying the proposed calculations. The discussion reflects varying interpretations of Bayesian principles and their application in this context.

Who May Find This Useful

This discussion may be of interest to individuals exploring Bayesian statistics, particularly in quality assurance contexts, as well as those seeking to understand the implications of false positives and negatives in probabilistic assessments.

Marione
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I have a set of 10000 elements to buy; I want to do some quality assurance activity in order to avoid to buy invalid elements. I have an automatic test that provides the probability of correctness of an element; moreover I know that the test gives a 20% of false positive and a 30 % of false negative.
1)Suppose that I have an element of which the test estimate a probability of correctness of 75%; which is the posterior probability?
I’m afraid I’m unable to apply the Bayesian theorem because I do not know the number of correct elements. My instinct tells me to count the frequency in the past in which an element was really correct when the estimated probability by the tool was less or equal then 75%; suppose that such frequency is 50%. Consequently, my instinct will say that an element of which the test estimate a probability of correctness of 75% has a posterior probability to be correct of 50%. Am I right?

2) Suppose that I cannot check all the elements by eye but I need to use the automatic test to prioritize the elements to check; suppose that I run the automatic test on all the elements and then I order the elements according to the probability of correctness provided by the test; suppose that I’ve analyzed all the elements with a probability of correctness lower 70%; how to calculate the probability that there is an incorrect elements in the remaining ones (i.e. the ones with a probability of correctness higher than 70%)?

Please help me!
 
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Can you tell me if this is a proper question picked up from somewhere or is any data missing here? If all correct, tell me if the answer for 1. is 15/16?
 
I don't understand your question.
 
A way to check your calculation is to count the overall defective percentage p[0]. Suppose p[0] = p. Then the probability of having a defective part conditional on a "pass score" p[0|1] can be found as:

p[0|1]p[1] = p[0,1] = p[1|0]p[0]
p[0|1] = p[1|0]p[0]/p[1]
p[0|1] = 0.2 p[0]/p[1]
p[0|1] = 0.2 p[0]/(1-p[0])
p[0|1] = 0.2 p/(1-p).

EnumaElish
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I would definitely have logged in as EnumaElish had PF administration awarded that account the privilege of posting replies, after I reset my e-mail address Tuesday, October 28, 2008.
 

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