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Suppose that X₁,..., Xn~ N(μ, 1) are independent and identically distributed Normal random variables with unknown parameter and variance 1, and that r = (₁,...,n) is data with sample mean given by a realisation of X₁,..., X₁. Suppose further that the prior distribution of the parameter μ has a Cauchy distribution with prior density function π(μ) = 1 ´π(1+µ²)* (a) Show that the posterior density function (μl) of the posterior distribution (ur) of the parameter u given the data x satisfies π(μ\x) x exp (nīu – ni) 1+μ² (b) Show that the maximum a priori (MAP) estimate μ (the mode of the posterior distribution (Alx)) of satisfies ²³-²+(1+2/n) µ − ñ = 0 (c) Show that the MAP estimate for large n. (d) If data x = (5.70, 4.81, 5.37, 3.75, 5.44, 5.02, 5.74, 3.62, 4.80, 5.32) with n = 10 data points is observed, then show that the MAP estimate of μ satisifes 4.9 ≤ ≤ 5.0.

Suppose that X₁,..., Xn~ N(μ, 1) are independent and identically distributed Normal random variables with unknown parameter and variance 1, and that r = (₁,...,n) is data with sample mean given by a realisation of X₁,..., X₁. Suppose further that the prior distribution of the parameter μ has a Cauchy distribution with prior density function π(μ) = 1 ´π(1+µ²)* (a) Show that the posterior density function (μl) of the posterior distribution (ur) of the parameter u given the data x satisfies π(μ\x) x exp (nīu – ni) 1+μ² (b) Show that the maximum a priori (MAP) estimate μ (the mode of the posterior distribution (Alx)) of satisfies ²³-²+(1+2/n) µ − ñ = 0 (c) Show that the MAP estimate for large n. (d) If data x = (5.70, 4.81, 5.37, 3.75, 5.44, 5.02, 5.74, 3.62, 4.80, 5.32) with n = 10 data points is observed, then show that the MAP estimate of μ satisifes 4.9 ≤ ≤ 5.0.

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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Suppose that X1,..., Xn ~ N(µ, 1) are independent and identically distributed Normal random variables with unknown parameter u and variance 1, and that r given by a realisation of X1,..., Xp. Suppose further that the prior distribution of the parameter u has a Cauchy distribution with prior density function (x1,..., xn) is data with sample mean 7 1 T(1+ µ?)* (a) Show that the posterior density function 1(4|x) of the posterior distribution (µ|x) of the parameter µ given the data x satisfies exp (пти — пр?) 1+ µ? T(µ|x) x (b) Show that the maximum a priori (MAP) estimate i (the mode of the posterior distribution (A|x)) of µ satisfies - T + (1+ 2/n) ũ – T = 0 (c) Show that the MAP estimate iT for large n. (d) If data r = (5.70, 4.81, 5.37,3.75, 5.44, 5.02, 5.74, 3.62, 4.80, 5.32) with n = 10 data points is observed, then show that the MAP estimate i of µu satisifes 4.9 <ñ< 5.0.
Transcribed Image Text:Suppose that X1,..., Xn ~ N(µ, 1) are independent and identically distributed Normal random variables with unknown parameter u and variance 1, and that r given by a realisation of X1,..., Xp. Suppose further that the prior distribution of the parameter u has a Cauchy distribution with prior density function (x1,..., xn) is data with sample mean 7 1 T(1+ µ?)* (a) Show that the posterior density function 1(4|x) of the posterior distribution (µ|x) of the parameter µ given the data x satisfies exp (пти — пр?) 1+ µ? T(µ|x) x (b) Show that the maximum a priori (MAP) estimate i (the mode of the posterior distribution (A|x)) of µ satisfies - T + (1+ 2/n) ũ – T = 0 (c) Show that the MAP estimate iT for large n. (d) If data r = (5.70, 4.81, 5.37,3.75, 5.44, 5.02, 5.74, 3.62, 4.80, 5.32) with n = 10 data points is observed, then show that the MAP estimate i of µu satisifes 4.9 <ñ< 5.0.
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