\frame{

\vspace{1cm}

\huge
\textbf{Chapter 3}

\vspace{1cm}

\Large
\textbf{General results for\\
multi-parameter problems}

}


\slidecolour
\mysection{General results for multi-parameter problems}

\begin{frame}
  \frametitle{Chapter \thesection. General results for multi-parameter
    problems}
\subsection{Different levels of  prior knowledge}

\begin{block}{{\thesubsection} Different levels of  prior knowledge}
\begin{itemize}
\item Substantial prior information:
  $\pi(\thetavec|\xvec)\sim\pi(\thetavec)$
\item Limited prior knowledge: use conjugate prior where possible and determine its parameter(s) by prior knowledge
\item Vague prior information: use conjugate prior where possible and make them diffuse
\item Prior ignorant -- use `flat' prior? Problem?
\begin{itemize}
\item [1.]Have $\boldsymbol{\theta} = (\theta_1, \theta_2, \ldots, \theta_p)^{T}$ and we are prior ignorant
\item [2.]Use $\pi(\boldsymbol{\theta}) = constant$?
\item [3.] Let $\phi_i = g_i(\theta), i=1, \ldots,p$ and $\boldsymbol{\phi} = (\phi_1, \phi_2, \ldots, \phi_p)^{T}$
\item [4.] Prior density for $\boldsymbol{\phi}$ is not constant!  
\item [5.] But if we are ignorant about $\boldsymbol{\theta}$ we must be ignorant about $\boldsymbol{\phi}$... 
\end{itemize}

\bigskip
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\vspace*{-3mm}
\begin{myexample}
Suppose $0<\theta_1<1$ and $0<\theta_2<1$.  If we are ignorant about $\thetavec = (\theta_1, \theta_2)^{T}$ then show that $\theta_1 \theta_2$ does not have constant prior density.  
\end{myexample}

\begin{soln}{}\end{soln}

\end{frame}


\begin{frame}
\vspace*{-3mm}
\begin{myexample}
Suppose we have a random sample from a $N(\mu, 1/\tau)$ distribution (with $\tau$ unknown).  Determine the Jeffreys prior for this model.

\vspace{0.5cm}

Hint: We have already seen that the likelihood function can be written as
\[
f(\xvec|\mu, \tau) = \left(\frac{\tau}{2\pi}\right)^{n/2}\exp\left[-\frac{n\tau}{2}\left\{s^2 + (\bar{x}-\mu)^2\right\}\right]
\]
where
\[
s^2 = \frac{1}{n} \sum_{i=1}^{n}(x_{i}-\bar{x})^2.
\]
\end{myexample}

\begin{soln}{}\end{soln}

\end{frame}

\subsection{Asymptotic posterior distribution}
\slidecolour
\begin{frame}
\vspace*{-5mm}
\frametitle{{\thesubsection} Asymptotic posterior distribution}
\begin{block}{}
\alert{
\[
p=1\text{ case}:\qquad
\sqrt{J(\hat\theta)}\left.\bigl(\theta-\hat\theta\bigr)
\right|\xvec\stackrel{\cal D} \longrightarrow N(0,1)
\quad\quad\text{as }n\to\infty
\]}

Suppose we have a statistical model $f(\xvec|\thetavec)$ for data
  $\xvece$, where $\thetavece$, together with a prior distribution
  $\pi(\thetavec)$ for $\thetavec$.  Then
\[
J(\thetavechat)^{1/2}(\thetavec-\thetavechat)|\xvec
\stackrel{\cal D} \longrightarrow N_p(\underline{0},\I_p)
\quad\quad\text{as }n\to\infty,
\]
where $\thetavechat$ is the likelihood mode, $\I_p$ is the
$p\times p$ identity matrix and $J(\thetavec)$ is the observed
information matrix, with $(i,j)^{th}$ element
\vspace*{-2mm}
\[
J_{ij}=-\frac{\partial^2}{\partial\theta_i\partial\theta_j}\,
\log f(\xvec|\thetavec),
\]
\vspace*{-2mm}
and $A^{1/2}$ denotes the square root matrix of $A$

\end{block}
\vspace*{-7mm}
\begin{block}{Comments}
When $n$ is large, $\thetavec\mid\xvec\sim N_p(\thetavechat,J(\thetavechat)^{-1})$ approximately.
\bigskip {\LARGE $\ldots$} \bigskip
\end{block}
\end{frame}

\begin{frame}
\vspace*{-3mm}
\begin{myexample}
Suppose we now have a random sample from a $N(\mu,1/\tau)$
\label{ex:asymprs}
distribution (with $\tau$ unknown). Determine the asymptotic
posterior distribution for~$(\mu,\tau)$. 

\bigskip
Hint: we have already seen that the likelihood function can be written
as
\[
f(\xvec|\mu,\tau)=\left(\frac{\tau}{2\pi}\right)^{n/2}
\exp\left[-\frac{n\tau}{2}
\left\{s^2+(\bar x-\mu)^2\right\}\right]
\]
where $s^2=\sum_{i=1}^n (x_i-\bar x)^2/n$
\end{myexample}

\begin{soln}{}\end{soln}

\end{frame}

\subsection{Learning objectives}
\slidecolour
\begin{frame}
\frametitle{{\thesubsection} Learning objectives}
\vspace*{-3mm}
\begin{block}{}
By the end of this chapter, you should be able to:
\begin{itemize}
\item[1.] describe the different levels of prior information;
  determine and use vague priors
\item[2.] determine the asymptotic posterior distribution when the data
  are a random sample from \textbf{any} distribution
\item[3.] explain the similarities and differences between the asymptotic
  posterior distribution and the asymptotic distribution of the
  maximum likelihood estimator
\end{itemize}
\end{block}
\end{frame}