# The Normal Model

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# The normal model

# Inference for the mean, conditional on the variance

# Joint inference for the mean and variance

posterior inference

and joint distribution can be

inverse-gamma distribution:

precision = $1/\sigma^2$

variance = $\sigma^2$

## posterior inference

# Bias, variance and mean squared error

# Prior specification based on expectations

- $t(y)=(y,y^2)$
- $\phi = (\theta/\sigma^2,-(2\sigma^2)^{-1})$
- $c(\phi)=\vert \phi_2\vert^{1/2}exp(\phi_1^2/(2\phi_2))$

a conjugate prior distribution

where $t_0=(E(Y), E(Y^2))$.