\section{Grove}
In this article,Grove adopted a model used by Lindley and Seheult and proposed a nonBayesian solution based on likelihood ratio.According to Grove:
\begin{quote}\qquad The measurements are normally distributed with known standard deviation $\sigma$ about a mean $\theta$. A measurement $x$, with normal error having known standard deviation $\sigma$, is made on the glass of  refractive index $\Theta_1$  at
the scene of the crime. Another measurement $y$, made on the glass found on 
the suspect, is also assumed to be normal but with mean $\theta_2$ and the 
same standard deviation as $x$. The refractive indices $\theta$ are assumed 
to be normally distributed with known mean $\mu$ and known standard deviation 
$\tau$. If $I$ is the event that the two pieces of glass come from the same 
source($\theta_1$ = $\theta_2$) and $\bar{I}$ the contrary event, Lindley 
suggests that the odds on identity should be mutliplied by the factor
\begin{equation}
\frac{p(x,y|I)}{p(x,y|\bar{I})} \label{1}
\end{equation}
\end{quote}
\begin{quote}\qquad Grove followed non Bayesian approach, so in discussing the respective roles of $\theta_1$ and $\theta_2$ he did not attribute
any probability distribution to an unknown parameter without special justification.So when considering ($\theta_1$ != $\theta_2$),$\bar{I}$ can be interpreted as saying that the fragments are present by chance entailing a random choice of value for $\theta_2$.The simplified likelihood ratio 
obtained from the Grove's derivation is:
\begin{equation}
\frac{\tau}{\sigma}\cdot e^{\big\{\frac{-(X-Y)^2}{4\sigma^{2}} + \frac{(Y-\mu^2)}{2\tau^2}\big\}} \label{2} 
\end{equation}
\end{quote}
\begin{quote}\qquad The Figure\ref{fig:groveidentities} demonstrates the implementation ot the grove equation for likelihood calculation in lisp.The function takes the
refractive indices of the samples from two places and compares them only when their standard deviation is same.Here in this fuction the $mu$ and $tau$ are the given parameters.The Figure\ref{fig:grovecomparision} demonstrates algorithhm to comapre two set of fragments here the likelihood ration is specified as a limit for the no of fragments to be considered as the proof in the investigation\end{quote}
\codelist{groveidentities}{pavan/grove.lisp}
\codelist{grovecomparision}{pavan/grovelr2.lisp}
\begin{quote}\qquad We are of course only concerned with the evidence about I and $\bar{I}$ so far as it has the bearing on the guilt or innocence of the suspect.
Grove also considered the Event of Guilty factor \b{G} in the calculation of likelihood ratio.Therefore the LR now becomes \begin{equation} p(X,Y|G)/p(X,Y|\bar{G})\end{equation}
Here in the expansion event \b{T}, that fragments were transferred from the broken window to the suspect and persisted until discovery and event \b{A},that 
the suspect came into contact with glass from other source.Here p(A/{G})=p(A/$\bar{G}$)=Pa and p(T/G)= Pt.The resulting expression is\newline

  \begin{equation} 
    \frac{P(X,Y,S|G)}{P(X,Y,S|\bar{G})} = 1+Pt\Big\{(\frac{1}{Pa}-1)\frac {p(X,Y|I)}{p(X,Y|\bar{I})}-1\Big\}\label{4}  
  \end{equation} 
\newline
 
\end{quote}
\begin{quote}\qquad The Figure \ref{fig:groveidentities} demonstrates the calculation of grove LR when the event $\bar{G}$ is considered.Here in the equation
Pa and Pt values are fixed an passed to the function directly.
\end{quote}
\begin{quote}\qquad
Figure \ref{img0} shows the results of running Grove on crime scene data.Note that the graph is concentrated only at the higer ends because the likelihood ration criteria is high resulting in the matching of fragments which have closer refractive indices.Figure \ref{img1} shows runnig Grove on survey data.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{pavan/test1.png}
\caption{crimescene Data}
\label{img0}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{pavan/test2.png}
\caption{Survey Data}
\label{img1}
\end{figure}
\end{quote}
