Consider the linear regression model Y = MB + E, where Y = (Y,..., Yn) is...

  

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Consider the linear regression model Y = MB + E, where Y = (Y₁,..., Yn) is the nx 1 response vector, M is the n × p model matrix with full column rank, ß is the p × 1 vector of unknown regression coefficients, and E is the nx 1 vector of random errors. Assume E~Nn (Onx1, ¹In), where > 0 is an unknown parameter, Onx1 is the n × 1 vector of 0's, and In is the n x n identity matrix. Write the vector of observed values of the Yi's as y = (V₁,..., yn). Assume the prior for (B,A) is given by g(B,A) x 2-¹1 (2> 0). Note that information in this paragraph should be used to answer the sub-questions below unless stated otherwise. (a) Write down the posterior PDF of (B,A), i.e., g(B,Aly), up to a proportionality constant. Please write it in a form that depends on y only through = (MTM)-¹MT y and RSS = y [In M(MTM)-¹MT]y. (b) Derive the marginal posterior PDF of 2, i.e., g(ly), up to a proportionality constant and identify the marginal posterior distribution of 2 (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of into a form that allows you to identify the marginal posterior distribution of λ. (c) Derive the marginal posterior PDF of ß, i.e., g (Bly), up to a proportionality constant and identify the marginal posterior distribution of ß (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of ß into a form that allows you to identify the marginal posterior distribution of B. (d) Based on the result you obtained in part (a), write down: 1. The full conditional PDF of ß up to a proportionality constant. 2. The full conditional PDF of up to a proportionality constant. Identify the full conditional distribution of ß and the full conditional distribution of 2 (state the name and parameters of each of the two full conditional distributions). Note: You need to write each of the two full conditional PDFs g (B2, y) and g(2[B, y) into a form that allows you to identify the distribution it represents. (e) Suppose Y(xº) |B, 2, y = Y(xº) |ß,a~N(m(xº)ß, λ-¹). Derive and identify the posterior predictive distribution y(xº)ly (please state the name of the distribution and give explicit expressions for its parameters). Consider the linear regression model Y = MB + E, where Y = (Y₁,..., Yn) is the nx 1 response vector, M is the n × p model matrix with full column rank, ß is the p × 1 vector of unknown regression coefficients, and E is the nx 1 vector of random errors. Assume E~Nn (Onx1, ¹In), where > 0 is an unknown parameter, Onx1 is the n × 1 vector of 0's, and In is the n x n identity matrix. Write the vector of observed values of the Yi's as y = (V₁,..., yn). Assume the prior for (B,A) is given by g(B,A) x 2-¹1 (2> 0). Note that information in this paragraph should be used to answer the sub-questions below unless stated otherwise. (a) Write down the posterior PDF of (B,A), i.e., g(B,Aly), up to a proportionality constant. Please write it in a form that depends on y only through = (MTM)-¹MT y and RSS = y [In M(MTM)-¹MT]y. (b) Derive the marginal posterior PDF of 2, i.e., g(ly), up to a proportionality constant and identify the marginal posterior distribution of 2 (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of into a form that allows you to identify the marginal posterior distribution of λ. (c) Derive the marginal posterior PDF of ß, i.e., g (Bly), up to a proportionality constant and identify the marginal posterior distribution of ß (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of ß into a form that allows you to identify the marginal posterior distribution of B. (d) Based on the result you obtained in part (a), write down: 1. The full conditional PDF of ß up to a proportionality constant. 2. The full conditional PDF of up to a proportionality constant. Identify the full conditional distribution of ß and the full conditional distribution of 2 (state the name and parameters of each of the two full conditional distributions). Note: You need to write each of the two full conditional PDFs g (B2, y) and g(2[B, y) into a form that allows you to identify the distribution it represents. (e) Suppose Y(xº) |B, 2, y = Y(xº) |ß,a~N(m(xº)ß, λ-¹). Derive and identify the posterior predictive distribution y(xº)ly (please state the name of the distribution and give explicit expressions for its parameters).

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a The posterior PDF of B given y is gB y gB x 2n22 x exp12yIn MMTMMTy BMTMMTy22 b ... View the full answer