

\subsection{\bf Enter Mr. Bayes}
\subsection{\bf for Conditional Probablities---I}

\begin{slide}
\begin{itemize}
\item Now what we really want is the probability distribution of an
estimate $\alpha$ of $\alpha_0$.  

\item We can formally get this by use of Bayes law:  

\item First extend $p_i \left( x_i \right) = \exp \left\lbrace -
\frac{1}{2\sigma^2_0} \left( x_i - \alpha \right)^2 \right\rbrace$ to
be defined for any value of $\alpha$; and not just for $\alpha = $ true
value $\alpha_0$.

\item What does this mean, as only one true $\alpha_0$?

\item Introduce:\ $P\left( \alpha \vert x_i , T \right) = $ Probability
that true length is $\alpha$ given measurements $x_i$ and prejudice $T$.

\item Then by Bayes---or rather a slight extension of it,
\begin{equation}
P \left( \alpha \vert x_i , T \right) = P \left( x_i \vert \alpha , 
T \right) \frac{P (\alpha \vert T)}{P (x_i \vert T)}
\end{equation}

\item which only differs from the ordinary Bayes Law by extra
conditional $\vert T$ in each probability.
\end{itemize}
\end{slide}

