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\begin{document}

\begin{center}\Large\bf 
        A Bayesian Outlier Detection Technique
\end{center}

\section{The OLS fit}

\begin{verbatim}
> summary(olsfit _ lm(LogRadon ~ Floor + Clay + Goodhue,data=radon))

Call: lm(formula = LogRadon ~ Floor + Clay + Goodhue, data = radon)
Residuals:
    Min      1Q   Median     3Q   Max 
 -1.653 -0.5572 -0.01608 0.5933 1.855

Coefficients:
              Value Std. Error t value Pr(>|t|) 
(Intercept)  1.9561  0.2158     9.0663  0.0000 
      Floor -0.3283  0.3155    -1.0409  0.3047 
       Clay -0.0798  0.3018    -0.2644  0.7929 
    Goodhue -0.0379  0.3041    -0.1248  0.9014 

Residual standard error: 0.7895 on 37 degrees of freedom
Multiple R-Squared: 0.0332 
F-statistic: 0.4235 on 3 and 37 degrees of freedom, the p-value is 0.7373 

Correlation of Coefficients:
        (Intercept)   Floor    Clay 
  Floor -0.2089                    
   Clay -0.6526     -0.1493        
Goodhue -0.6762     -0.0114  0.4868
\end{verbatim}
\begin{figure}
\centerline{\psfig{figure=olsdiag.ps,height=3.5in}}
\caption{Standard Splus diagnostics for the OLS fit.  Note that
observations X, Y and Z, have maximum infleunce and Z and X seem to be
outliers.}\label{olsdiag}
\end{figure}

\section{The model}

For each $i$, let
\[
	z_i ~ = ~ \left\{\begin{array}{ll}
			\mbox{1, if obs $i$ is an outlier;} \\
			\mbox{2, if obs $i$ is not an outlier.}
			\end{array}\right. 
\]
Then our model is
\begin{eqnarray*}
	y_i & \sim &
		\left\{\begin{array}{ll}
		\mbox{$\mb x_i\tp\mb\beta + 
			\sqrt{(\nu-2)/\nu}\cdot\sigma\cdot t_{\nu}$,
			if $z_i=1$;} \\
		\mbox{$\mb x_i\tp\mb\beta + \sigma \cdot N(0,1)$, if $z_i=2$.} 
			\end{array}\right. \\
	z_i & = & \left\{\begin{array}{ll}
			\mbox{1, with probability $p_1$;} \\
			\mbox{2, with probability $p_2$.}
			\end{array}\right. \\
	\mb\beta & \sim & \mbox{flat} \\
	\log(\sigma) & \sim & \mbox{flat} \\
	p_1 &\sim & Beta(a,b) \\
	p_2 & = & 1-p_1
\end{eqnarray*}

\section{The BUGS code}

\section{The output}

\end{document}









