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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.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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.
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