The Normal Model
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The normal model
\[p(y\mid \theta,\sigma^2)=\frac{1}{\sqrt{(2\pi \sigma^2)}}e^{-\frac{1}{2}(\frac{y-\theta}{\sigma})^2}, -\infty<y<\infty\]Inference for the mean, conditional on the variance
Joint inference for the mean and variance
posterior inference
\[p(\theta,\sigma^2\mid y_1,\ldots,y_n)=p(y_1,\ldots,y_n\mid \theta,\sigma^2)p(\theta,\sigma^2)/p(y_1,\ldots,y_n)\]and joint distribution can be
\[p(\theta,\sigma^2)=p(\theta\mid\sigma^2)p(\sigma^2)\]inverse-gamma distribution:
precision = $1/\sigma^2$
\[1/\sigma^2\sim gamma(a,b)\]variance = $\sigma^2$
\[\sigma\sim inverse-gamma(a,b)\]posterior inference
\(\begin{align} 1/\sigma^2 & \sim gamma(v_0/2, v_0\sigma^2_0/2)\\ \theta\mid \sigma^2 &\sim normal(\mu_0,\sigma/\kappa_0)\\ Y_1,\ldots,Y_n\mid \theta,\sigma^2 &\sim i.i.d normal(\theta, \sigma^2) \end{align}\)
Bias, variance and mean squared error
\[MSE(\hat\theta\mid\theta_0)=Var(\hat\theta\mid\theta_0)+Bias^2(\hat\theta\mid\theta_0)\]Prior specification based on expectations
\[p(y\mid\phi) = h(y)c(\phi)exp(\phi^Tt(y))\]- $t(y)=(y,y^2)$
- $\phi = (\theta/\sigma^2,-(2\sigma^2)^{-1})$
- $c(\phi)=\vert \phi_2\vert^{1/2}exp(\phi_1^2/(2\phi_2))$
a conjugate prior distribution \(p(\phi\mid n_0,t_0)\propto c(\phi)^{n_0}exp(n_0t_0^T\phi)\)
where $t_0=(E(Y), E(Y^2))$.