a) There are 6 balls in the bowl, of which 0 is white and the rest black. Person A has no knowledge of how to make balls the colors are determined, so according to his preconception all options (0,1,2,3,4,5,6) for 0 are equal likely. The person B picks up the ball from the bowl at random, returning it to the bowl and says the color is black. B repeats this one more time and says the color is black again. Person A does not know person B. Person A tries to figure out the number of white balls in the bowsl using Bayesian inference. Express by listing the values or, if desired, also by the formula i) the prior distribution of from the perspective of person A, ii) likelihood function the person A uses, iii) the person A' posterior distribution for 0. What is the most probable value of from the perspective of person A after making the observations? b) Continuation: The person C is listening the same experiment, without seeing the bowl either. Person C happens to know person B and knows that person B has a habit of playing tricks sometimes and that B is telling with a probability of 1/8 (independently) the opposite color that they actually see. So B might say the ball is black even though it is white with a probability of 1/8. With probability 7/8 the person B will tell the true color of the observed ball. Express the values by listing and, if desired, also by the formula i) the person C's prior distribution for e, ii) the likelihood function form the perspective of person C, iii) person C' posterior distribution for e. What is the most probable value of 0 from the perspective of person C after making observations? c) Continuation: Calculate the posterior mean value from the perspective of person C. Also calculate the posterior probability of the hypothesis {0 ≤ 2} from the point of view of person C.
a) There are 6 balls in the bowl, of which 0 is white and the rest black. Person A has no knowledge of how to make balls the colors are determined, so according to his preconception all options (0,1,2,3,4,5,6) for 0 are equal likely. The person B picks up the ball from the bowl at random, returning it to the bowl and says the color is black. B repeats this one more time and says the color is black again. Person A does not know person B. Person A tries to figure out the number of white balls in the bowsl using Bayesian inference. Express by listing the values or, if desired, also by the formula i) the prior distribution of from the perspective of person A, ii) likelihood function the person A uses, iii) the person A' posterior distribution for 0. What is the most probable value of from the perspective of person A after making the observations? b) Continuation: The person C is listening the same experiment, without seeing the bowl either. Person C happens to know person B and knows that person B has a habit of playing tricks sometimes and that B is telling with a probability of 1/8 (independently) the opposite color that they actually see. So B might say the ball is black even though it is white with a probability of 1/8. With probability 7/8 the person B will tell the true color of the observed ball. Express the values by listing and, if desired, also by the formula i) the person C's prior distribution for e, ii) the likelihood function form the perspective of person C, iii) person C' posterior distribution for e. What is the most probable value of 0 from the perspective of person C after making observations? c) Continuation: Calculate the posterior mean value from the perspective of person C. Also calculate the posterior probability of the hypothesis {0 ≤ 2} from the point of view of person C.
Chapter9: Sequences, Probability And Counting Theory
Section9.5: Counting Principles
Problem 40SE: A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in...
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Transcribed Image Text:a)
There are 6 balls in the bowl, of which 0 is white and the rest black. Person A has
no knowledge of how to make balls the colors are determined, so according to his
preconception all options (0,1,2,3,4,5,6) for 0 are equal likely.
The person B picks up the ball from the bowl at random, returning it to the bowl
and says the color is black. B repeats this one more time and says the color is black
again. Person A does not know person B. Person A tries to figure out the number
of white balls in the bowsl using Bayesian inference.
Express by listing the values or, if desired, also by the formula i) the prior distribution
of from the perspective of person A, ii) likelihood function the person A uses,
iii) the person A' posterior distribution for 0. What is the most probable value of
from the perspective of person A after making the observations?
b) Continuation: The person C is listening the same experiment, without seeing
the bowl either. Person C happens to know person B and knows that person B
has a habit of playing tricks sometimes and that B is telling with a probability
of 1/8 (independently) the opposite color that they actually see. So B might
say the ball is black even though it is white with a probability of 1/8. With
probability 7/8 the person B will tell the true color of the observed ball.
Express the values by listing and, if desired, also by the formula i) the person
C's prior distribution for e, ii) the likelihood function form the perspective of
person C, iii) person C' posterior distribution for e. What is the most probable
value of 0 from the perspective of person C after making observations?
c) Continuation: Calculate the posterior mean value from the perspective of
person C. Also calculate the posterior probability of the hypothesis {0 ≤ 2}
from the point of view of person C.
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