# Given the data (tau=left{x_{1}, ldots, x_{n} ight}), suppose that we use the likelihood ((X mid boldsymbol{theta}) sim

## Question:

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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Related Book For

## 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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