In this section $ x_1,x_2,\cdots,x_n $ are iid observations from the Normal distribution with density:
$ f(x\mid\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp\{-\frac{(x-\mu)^2}{2\sigma^2}\},\,-\infty<x<\infty\,\,(-\infty<\mu<\infty,\sigma>0). $
In the parameterization above $ \mu $ and $ \sigma $ represent the mean and standard deviation. Other parameterizations are in terms of the variance $ \sigma^2 $ or the precision $ \tau=1/\sigma^2 $
Case 1: Mean and standard deviation unknown Edit
Case 1: Recommended prior |
---|
$ \pi(\mu,\sigma)\propto 1/\sigma $ |
In other parameterizations |
$ \pi(\mu,\sigma^2)\propto 1/\sigma^2 $ |
$ \pi(\mu,\tau)\propto 1/\tau $ |
Formal endorsements Edit
^{[1]} showed that the recommended prior is the reference prior for any ordering of the parameters and hence it is the overall recommended prior in ^{[2]}. Furthermore, this is the Jeffreys' independent prior.
Posterior propriety Edit
The posterior distribution associated with this prior is proper if $ n\ge 2 $(reference here).
Other properties Edit
Other important properties of this prior have to be with invariance arguments. In this distribution, $ (\mu,\sigma) $ are location-scale invariant parameters so the ....
Alternatives Edit
The multivariate Jeffreys prior is $ \pi(\mu,\sigma)\propto 1/\sigma^2 $, but as argued in () this is much less appealing.
Other facts Edit
This prior is a limiting case of proper priors of the form...
Case 2: Mean unknown; standard deviation known Edit
Case 2: Recommended prior |
---|
$ \pi(\mu)\propto 1 $ |
Formal endorsements Edit
This prior is the reference prior ^{[1]} for the ordering xxxx and it is the overall recommended prior in Berger, Bernardo and Sun (200?). Furthermore, this is the Jeffreys independent prior.
Posterior propriety Edit
The posterior distribution associated with this prior is proper if $ n\ge 1 $(reference here).
Other properties Edit
Other important properties of this prior have to be with invariance arguments. In this distribution, $ (\mu,\sigma) $ are location-scale invariant parameters so the ....
This prior is a limiting case of proper priors of the form...
Case 3: Standard deviation unknown; mean known Edit
Suppose $ \sigma $ is unknown.
Case 3: Recommended prior |
---|
$ \pi(\sigma)\propto 1/\sigma $ |
In other parameterizations |
$ \pi(\sigma^2)\propto 1/\sigma^2 $ |
$ \pi(\tau)\propto 1/\tau $ |
Formal endorsements Edit
This prior is the reference prior (for the orderings ...) and it is the overall recommended prior in Berger, Bernardo and Sun (200?). Furthermore, this is the Jeffreys independent prior.
Posterior propriety Edit
the posterior distribution associated with this prior is proper if $ n\ge 2 $(reference ).
Other properties Edit
Other important properties of this prior have to be with invariance arguments. In this distribution, $ (\mu,\sigma) $ are location-scale invariant parameters so the ....
This prior is a limiting case of proper priors of the form...
ReferencesEdit
- ↑ ^{1.0} ^{1.1} Bernardo, J. and Smith, A. (1994), Bayesian Theory, John Wiley and Sons, London.
- ↑ Berger, J., Bernardo, J., and Sun, D. (2015), Overall objective priors, Bayesian Analysis, 10, 189-221.