4.7 Exercises: Chapter 4

1. Write in the canonical form the distribution of the Bernoulli example, and find the mean and variance of the sufficient statistic.

2. Given a random sample \(\mathbf{y}=[\mathbf{y}_1,\mathbf{y}_2,\dots,\mathbf{y}_N]^{\top}\) from a binomial distribution where the number of trials (\(n\)) is known. Show that \(p(\mathbf{y}|\theta)\) is in the exponential family, and find the posterior distribution, the marginal likelihood and the predictive distribution of the binomial-beta model assuming the number of trials is known.

3. Given a random sample \(\mathbf{y}=[y_1,y_2,\dots,y_N]^{\top}\) from a exponential distribution. Show that \(p(\mathbf{y}|\alpha,\beta)\) is in the exponential family, and find the posterior distribution, marginal likelihood and predictive distribution of the exponential-gamma model.

4. Find the marginal likelihood in the normal/inverse-Wishart model.

5. Find the posterior predictive distribution in the normal/inverse-Wishart model.

6. Show that in the linear regression model \(\beta_n^{\top}({\bf{B}}_n^{-1}-{\bf{B}}_n^{-1}{\bf{M}}^{-1}{\bf{B}}_n^{-1})\beta_n={\bf{\beta}}_{**}^{\top}{\bf{C}}{\bf{\beta}}_{**}\) and \(\beta_{**}={\mathbf{X}}_0\beta_n\).

7. Show that \(({\bf{Y}}-{\bf{X}}{\bf{B}})^{\top}({\bf{Y}}-{\bf{X}}{\bf{B}})={\bf{S}}+({\bf{B}}-\widehat{\bf{B}})^{\top}{\bf{X}}^{\top}{\bf{X}}({\bf{B}}-\widehat{\bf{B}})\) where \({\bf{S}}= ({\bf{Y}}-{\bf{X}}\widehat{\bf{B}})^{\top}({\bf{Y}}-{\bf{X}}\widehat{\bf{B}})\), \(\widehat{\bf{B}}= ({\bf{X}}^{\top}{\bf{X}})^{-1}{\bf{X}}^{\top}{\bf{Y}}\).