What is Bayes rule: Definition and 15 Discussions

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule; recently Bayes–Price theorem), named after the Reverend Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole.
One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesian statistics.

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  1. B

    Mixed random variables problem

    I got (a) and (b) but I'm still working on (c). The solutions can be found here for your reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/lecture-9/MIT6_041SCF13_assn05_sol.pdf. But...
  2. C

    I Optimizing Defective Rates with Bayes' Formula

    Hello! I'm sitting with a problem that is causing me some troubles.. First part is using Bayes formula. We have 3 companies that produce some apparatus. Each company has some defective percentage. Company Produced (%) Defective (%) A 45 3 B 25 6 C 30 5 1) Suppose we pick up a...
  3. P-values Broke Scientific Statistics—Can We Fix Them?

    P-values Broke Scientific Statistics—Can We Fix Them?

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  4. Vital

    I Getting to Grips with Bayes' Formula: Exploring the Problem & Questions

    Hello! I am trying to get to grips with the Bayes' formula by developing an intuition about the formula itself, and on how to use it, and how to interpret. Please, take a problem, and my questions written within them - I will highlight my questions and will post them as I add the information...
  5. Z

    Solve Bayes' Rule Question: Probability of Rain Tomorrow

    Hi, I'm not sure whether my understanding of this question is correct: An app predicts rain tomorrow. Recently, it has rained only 73 days each year. When it actually rains, the app correctly forecasts rain 70% of the time. When it does not rain, it incorrectly forecasts rain 30% of the time...
  6. defaultusername

    Monty Hall Problem: Probability Calculation for Switching Doors

    Homework Statement Suppose you’re on a game show and you’re given the choice of three doors. Behind one is a car, behind the others are goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number 3 which has a goat. He says to you, “Do...
  7. N

    Is P(A,B|C) = P(A|C) P(B|C), if P(A,B) = P(A)P(B)?

    As stated in my subject line, I know that P(A|B) = P(A) and P(B|A) = P(B), i.e. A and B are separable as P(A,B) = P(A) P(B). I strongly suspect that this holds with a conditional added, but I can't find a way to formally prove it... can anyone prove this in a couple of lines via Bayes' rules...
  8. W

    Prob/Stats Easy text on conditional probability and Bayes theory

    Hi, I am teaching a machine learning course and the students have very poor knowledge about conditional probability, Bayes rule etc. Most students have done their undergraduates years ago and I guess their educational background has not been that good. Last lecture was on Naive Bayes...
  9. C

    Probability Conditional Expectation

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  10. E

    Bayes rule and joint probability

    I try to understand the following equation but I can not. I have basic knowledge about Bayes rule and joint probability. How can we produce this result? I would appreciate any help. p(R|q,x)/p(NR|q,x)=[p(R|q) / p(NR|q) ] * [p(x|R,q) / p(x|NR,q)]
  11. A

    The product rule and Bayes rule

    Is the rule: P(AB I) = P(BA I) (which is used to derive Bayes rule) an axiom for probability? And if so, do you guys find it intuitive that it should hold. For instance consider a box with green and red beads. Do you think it is strictly obvious that the probability of getting red-green is...
  12. I

    Calculating Probability of Coin Selection Using Bayes Rule

    Homework Statement I am just wanting to get a good starting point... not an answer to the question. Question: A desk contains three drawers. Drawer 1 has two gold coins. Drawer 3 has one gold coin and one silver coin. Drawer 3 has two silver coins. I randomly choose a drawer and then randomly...
  13. T

    Conditional Probability without Bayes rule

    Say I want to find Pr(A|B). Usually I would just use bayes rule, but some textbooks just assume that A|B works and then just multiply it by P(B). For example in my book, Pr(X>2000)=.4. X and Y are unif distributed from 1000 to 5000. They are independent. Find Pr(X+Y>8000|X>2000). Well they just...
  14. V

    Bayes Rule Probability Problem

    Homework Statement Event B is cow has BSE Event T is the test for BSE is positive P(B) = 1.3*10^-5 P(T|B) = .70 probability that the test is positive, given that the cow has BSE P(T|Bcc) = .10 probability that the test is positive given that the cow does not have BSE Find P(B|T) and...
  15. A

    Bayes rule using higher order prior probability

    Hi I am asking, if I am trying to make inference using Bayes rule based on a prior probability that is a random variable by itself; is it sufficient to use the expected value of such probability or there are other details. Thanks in advance.
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