# When using Jeffrey's prior for Normal model, what is $p_J(\theta, \sigma^{2} | y_{1}, ..., y_{n})$ supposed to be?

I'm reading A First Course in Bayesian Statistical Methods by P. Hoff where he is using Jeffrey's prior (J) and Unit information prior (U) for Normal model. For example we can derive Jeffrey's prior for Normal model and get $p_{J}(\theta, \sigma^2) = (\sigma^2)^{-3/2}$ and then obtain a posterior $p(\theta, \sigma^2 | y_{1}, ..., y_{n})$ as the product of prior and likelihood. I don't understand what he means by probability density with subscript J, for example: $p_{J}(\theta, \sigma^{2} | y_{1}, ..., y_{n})$ or $p_{J}(\theta | \sigma^2, y_{1}, ..., y_{n})$. Can someone please explain?

• In the context of the problem, it means the posterior pdf associated with Jeffrey's prior. i.e. it is $p_J(\theta,\sigma^2|y_1,...,y_n)\propto p_J(\theta,\sigma^2)p(y_1,...,y_n|\theta,\sigma^2)$ Apr 24 '17 at 13:30
• Thank you for your answer, so it is only the posterior we obtain as the product of Jeffrey's prior and likelihood? And what is $p_{J}(\theta | \sigma^2, y_{1}, ..., y_{n})$ and $p_{J}(\sigma^2 | y_{1}, ..., y_{n})$ supposed to be and how to derive it? Apr 24 '17 at 14:00
• In general the posterior is the product of the prior and the likelihood. The subscript J is only to indicate that in the particular example the Jeffrey's prior is used. Consequently the $p_{J}(\theta|\ldots)$ and $p_{J}(\sigma^2|\ldots)$ are the full conditional for the parameters $\theta,\sigma^2$ which are obtained from the full posterior under the Jeffreys prior. Apr 24 '17 at 14:16

## 1 Answer

In general the posterior is the product of the prior and the likelihood. The subscript J is only to indicate that in the particular example the Jeffrey's prior is used. Consequently the

$p_{J}(\theta|\ldots)$ and $p_{J}(\sigma^2|\ldots)$

are the full conditional for the parameters $\theta,\sigma^2$ which are obtained from the full posterior under the Jeffreys prior. You derive them from the full posterior as the distribution depending only on the parameter of interest up to a constant of proportionality.

For example suppose that we have a posterior of the form

$p(\mu,\lambda|\ldots) \propto \lambda^{\frac{1}{2}}e^{-\frac{\lambda}{2}(x-\mu)^2}.$

Then the full conditional for $\lambda$ is

$p(\lambda|\ldots) \propto \lambda^{\frac{1}{2}}e^{-\frac{\lambda}{2}(x-\mu)^2},$ which is a $Ga\left(\frac{3}{2},\frac{1}{2}(x-\mu)^2\right),$

and for $\mu$

$p(\mu|\ldots) \propto e^{-\frac{\lambda}{2}(x-\mu)^2},$ which after some algebra is a normal distribution.

• Thank you for your answer. I understand now. For my example, Normal model with Jeffrey's prior, I derived full conditionals $p_{J}(\theta | \sigma^{2}, y_{1}, ..., y_{n}) \propto \texttt{Normal}(\bar{y}, \sigma^2/n)$ and $p_{J}(\sigma^{2} | y_{1}, ..., y_{n}) \propto \texttt{InverseGamma}(n/2, \sum_{i=1}^{n}(y_{i} - \bar{y})^{2}/2)$. Apr 24 '17 at 16:16
• No problem at all. I am glad i helped! Apr 25 '17 at 0:10