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\subsection{Distinguishing tests}  There are more interesting cases, on which some of the test statistics fail but others success.  \begin{itemize}  \item \verb|Bi_G500_far_bias| Olga??? LRT  \item The mock data \verb|Bi_G500_close_bias| represent the case that LRT is better than EMM. The use of the prior information that all modes have Guassian shape by LRT helps it to give stronger inference than EMM with no input prior. It gives us an interesting implementation that, if one has already known the intrinsic shape of each modes, LRT can absorb this information and make stronger inference than non-parametric statistics.  \item \verb|Uni_U500| is a mock data sampled from standard uniform distribution $U(0,1)$. Using GMM to fit the data, one can see that the more the modes it uses, the better the fit. All the GMM-based statistics reject one modes hypothesis and favor multi modes. One can also test that these methods require infinite number of modes in order to fit the uniform distribution, which is definitely wrong. Instead, EMM gives very large p-value, indicating a unimodal distribution. This imply that, if the Guassianality is invalid, all GMM-based methods can mislead the result, and a model-independent method is required.  \end{itemize}