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Given the data \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\), suppose that we use the likelihood \((X \mid \boldsymbol{\theta}) \sim \mathscr{N}\left(\mu, \sigma^{2}\right)\) with parameter \(\boldsymbol{\theta}=\left(\mu, \sigma^{2}\right)^{\top}\) and wish to compare the following two nested models.

(a) Model \(p=1\), where \(\sigma^{2}=\sigma_{0}^{2}\) is known and this is incorporated via the prior

\[ g(\boldsymbol{\theta} \mid p=1)=g\left(\mu \mid \sigma^{2}, p=1\right) g\left(\sigma^{2} \mid p=1\right)=\frac{1}{\sqrt{2 \pi \sigma}} \mathrm{e}^{-\frac{\left(\mu-x_{0}\right)^{2}}{2 \sigma^{2}}} \times \delta\left(\sigma^{2}-\sigma_{0}^{2}\right) \]

(b) Model \(p=2\), where both mean and variance are unknown with prior

\[ g(\boldsymbol{\theta} \mid p=2)=g\left(\mu \mid \sigma^{2}\right) g\left(\sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma}} \mathrm{e}^{-\frac{\left(\mu-x_{0}\right)^{2}}{2 \sigma^{2}}} \times \frac{b^{t}\left(\sigma^{2}\right)^{-t-1} \mathrm{e}^{-b / \sigma^{2}}}{\Gamma(t)} \]

Show that the prior \(g(\boldsymbol{\theta} \mid p=1)\) can be viewed as the limit of the prior \(g(\boldsymbol{\theta} \mid p=2)\) when \(t \rightarrow\) \(\infty\) and \(b=t \sigma_{0}^{2}\) Hence, conclude that

\[ g(\tau \mid p=1)=\lim _{\substack{t \rightarrow \infty \\ b=t \sigma_{0}^{2}}} g(\tau \mid p=2) \]

and use this result to calculate \(B_{112}\). Check that the formula for \(B_{1 \mid 2}\) agrees with the Savage-Dickey density ratio:

\[ \frac{g(\tau \mid p=1)}{g(\tau \mid p=2)}=\frac{g\left(\sigma^{2}=\sigma_{0}^{2} \mid \tau\right)}{g\left(\sigma^{2}=\sigma_{0}^{2}\right)} \]

where \(g\left(\sigma^{2} \mid \tau\right)\) and \(g\left(\sigma^{2}\right)\) are the posterior and prior, respectively, under model \(p=2\).

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Data Science And Machine Learning Mathematical And Statistical Methods

ISBN: 9781118710852

1st Edition

Authors: Dirk P. Kroese, Thomas Taimre, Radislav Vaisman, Zdravko Botev

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