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Question 4 (a) Describe the estimator Ô for a quantity 0 (which you should also determine) that would be obtained by the following R code N=10000 sample rnorm (N) sum (sample <=2)/N Let y = (y,,y) be independent and identically distributed data from N(0, 0²), where σ² is known and is unknown. We want to estimate 0. We assume a prior N(μ, t²) for 0, where both μ and 7 are known. The posterior density of 0 is N(μ₁, σ²). (b) Show how you would use the MH algorithm to obtain samples from the posterior density of 0, N(μ₁, σ²) (c) Describe the following MCMC implementation issues and how they may be dealt with in practice (i) Starting values. (ii) Dependence of the iterations.

Question 4 (a) Describe the estimator Ô for a quantity 0 (which you should also determine) that would be obtained by the following R code N=10000 sample rnorm (N) sum (sample <=2)/N Let y = (y,,y) be independent and identically distributed data from N(0, 0²), where σ² is known and is unknown. We want to estimate 0. We assume a prior N(μ, t²) for 0, where both μ and 7 are known. The posterior density of 0 is N(μ₁, σ²). (b) Show how you would use the MH algorithm to obtain samples from the posterior density of 0, N(μ₁, σ²) (c) Describe the following MCMC implementation issues and how they may be dealt with in practice (i) Starting values. (ii) Dependence of the iterations.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 4 (a) Describe the estimator Ô for a quantity 0 (which you should also determine) that would be obtained by the following R code N=10000 sample rnorm (N) sum (sample <=2)/N Let y = (y,,y) be independent and identically distributed data from N(0, 0²), where σ² is known and is unknown. We want to estimate 0. We assume a prior N(μ, t²) for 0, where both μ and 7 are known. The posterior density of 0 is N(μ₁, σ²). (b) Show how you would use the MH algorithm to obtain samples from the posterior density of 0, N(μ₁, σ²) (c) Describe the following MCMC implementation issues and how they may be dealt with in practice (i) Starting values. (ii) Dependence of the iterations.
Transcribed Image Text:Question 4 (a) Describe the estimator Ô for a quantity 0 (which you should also determine) that would be obtained by the following R code N=10000 sample rnorm (N) sum (sample <=2)/N Let y = (y,,y) be independent and identically distributed data from N(0, 0²), where σ² is known and is unknown. We want to estimate 0. We assume a prior N(μ, t²) for 0, where both μ and 7 are known. The posterior density of 0 is N(μ₁, σ²). (b) Show how you would use the MH algorithm to obtain samples from the posterior density of 0, N(μ₁, σ²) (c) Describe the following MCMC implementation issues and how they may be dealt with in practice (i) Starting values. (ii) Dependence of the iterations.
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