Homework Help is Here – Start Your Trial Now!
Math
Statistics
NOTE. For positive integers a and b, you may assume that (а- 1)! (ь — 1)! (а +b—1)! za-1(1 – 2)*-1 dz = 1. Consider a 6-sided dice in which a dice roll has a probability 0 of showing a 6 and a probability (1 – 0) of showing some other number (1, 2, 3, 4, 5), where 0 < 0 < 1. For a series of dice rolls, let x1,.., In be the number of dice rolls between successive 6s, so the dice roll sequence 1,6, 1,3, 5, 2, 6, ... would have r1 = 2, x2 = 5 and so on, with sample mean I = (r+...+In). %3D (a) Show that the likelihood function L(0; x) for the parameter 0 given = (x1,..., In) is L(0; x) = 0" (1 – 0)"(z-1), and show 1 the data x that the Maximum Likelihood (ML) estimate 0 of 0 is = : (b) For a uniform prior distribution, show that the posterior density function 1(0|x) of the posterior distribution (0|x) is given by (nF + 1)! n! (n(T – 1))! 7(0|x) = O" (1 – 0)n(7-1). n+1 (c) Show that mean of the posterior distribution is E((0|r)) = %3D nI + 2 (d) Find the marimum a posteriori (MAP) estimate of 0, that is to say the value of 0 maximising ¤(0|x). (e) Find the ML estimate ô of 0, the MAP estimate of 0 and the posterior mean E((0|x)) of the posterior distribution (0|x) for the data x = (2,1, 5, 9, 12, 3, 1, 1, 8, 4) with n = 10 data points.

NOTE. For positive integers a and b, you may assume that (а- 1)! (ь — 1)! (а +b—1)! za-1(1 – 2)*-1 dz = 1. Consider a 6-sided dice in which a dice roll has a probability 0 of showing a 6 and a probability (1 – 0) of showing some other number (1, 2, 3, 4, 5), where 0 < 0 < 1. For a series of dice rolls, let x1,.., In be the number of dice rolls between successive 6s, so the dice roll sequence 1,6, 1,3, 5, 2, 6, ... would have r1 = 2, x2 = 5 and so on, with sample mean I = (r+...+In). %3D (a) Show that the likelihood function L(0; x) for the parameter 0 given = (x1,..., In) is L(0; x) = 0" (1 – 0)"(z-1), and show 1 the data x that the Maximum Likelihood (ML) estimate 0 of 0 is = : (b) For a uniform prior distribution, show that the posterior density function 1(0|x) of the posterior distribution (0|x) is given by (nF + 1)! n! (n(T – 1))! 7(0|x) = O" (1 – 0)n(7-1). n+1 (c) Show that mean of the posterior distribution is E((0|r)) = %3D nI + 2 (d) Find the marimum a posteriori (MAP) estimate of 0, that is to say the value of 0 maximising ¤(0|x). (e) Find the ML estimate ô of 0, the MAP estimate of 0 and the posterior mean E((0|x)) of the posterior distribution (0|x) for the data x = (2,1, 5, 9, 12, 3, 1, 1, 8, 4) with n = 10 data points.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 56EQ
See similar textbooks
icon
Related questions
Question
100%

Bayesian Inference

NOTE. For positive integers a and b, you may assume that | 2a-'(1 – 2)*-1 dz = (а — 1)! (b — 1)! (a + b – 1)! 1. Consider a 6-sided dice in which a dice roll has a probability 0 of showing a 6 and a probability (1 – 0) of showing some other number (1, 2, 3, 4, 5), where 0 < 0 < 1. For a series of dice rolls, let x1,..., Tn be the number of dice rolls between successive 6s, so the dice roll sequence 1, 6, 1, 3, 5, 2, 6, ... would have xı = 2, x2 = 5 and so on, with sample %3D (*1 +...+ xn). mean x (a) Show that the likelihood function L(0; x) for the parameter 0 given (*1,..., xn) is L(0; x) = 0" (1 – 0)"(u-1), and show the data x = 1 that the Maximum Likelihood (ML) estimate 0 of 0 is 0 (b) For a uniform prior distribution, show that the posterior density function 7(0|x) of the posterior distribution (0|x) is given by (nF + 1)! n! (n(T – 1))! T(0|x) = O" (1 – 0)n(z-1). n +1 (c) Show that mean of the posterior distribution is E((0\x)) : nữ + 2 (d) Find the marimum a posteriori (MAP) estimate ô of 0, that is to say the value of 0 maximising 1(0|x). (e) Find the ML estimate ô of 0, the MAP estimate of 0 and the posterior mean E((0|x)) of the posterior distribution (0|x) for the data x = (2,1,5, 9, 12, 3, 1, 1, 8, 4) with n = 10 data points.
Transcribed Image Text:NOTE. For positive integers a and b, you may assume that | 2a-'(1 – 2)*-1 dz = (а — 1)! (b — 1)! (a + b – 1)! 1. Consider a 6-sided dice in which a dice roll has a probability 0 of showing a 6 and a probability (1 – 0) of showing some other number (1, 2, 3, 4, 5), where 0 < 0 < 1. For a series of dice rolls, let x1,..., Tn be the number of dice rolls between successive 6s, so the dice roll sequence 1, 6, 1, 3, 5, 2, 6, ... would have xı = 2, x2 = 5 and so on, with sample %3D (*1 +...+ xn). mean x (a) Show that the likelihood function L(0; x) for the parameter 0 given (*1,..., xn) is L(0; x) = 0" (1 – 0)"(u-1), and show the data x = 1 that the Maximum Likelihood (ML) estimate 0 of 0 is 0 (b) For a uniform prior distribution, show that the posterior density function 7(0|x) of the posterior distribution (0|x) is given by (nF + 1)! n! (n(T – 1))! T(0|x) = O" (1 – 0)n(z-1). n +1 (c) Show that mean of the posterior distribution is E((0\x)) : nữ + 2 (d) Find the marimum a posteriori (MAP) estimate ô of 0, that is to say the value of 0 maximising 1(0|x). (e) Find the ML estimate ô of 0, the MAP estimate of 0 and the posterior mean E((0|x)) of the posterior distribution (0|x) for the data x = (2,1,5, 9, 12, 3, 1, 1, 8, 4) with n = 10 data points.
Expert Solution
steps

Step by step

Solved in 6 steps with 17 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS