Hi,
I have built a Gaussian Mixture model to check the likelihood of each point and then identify the outliers by plotting the log likelihood as in the graph attached. In this particular case, the points below -2 can be considered outlier by visualizing how far they exist from the rest of points. The challenge is that I have hundreds of such cases and would like to plot and check manually the outliers. Is there a statistical way to determine the cut-off point (like -2 in this case) so I can write a program to avoid manual work..
It seems you want to arbitrarily label points with relatively low likelihood under your model as "outliers". I don't think there is any "objective" rule to justify an particular cut-off. As a heuristic you may specify what proportion of the values you can accept being labeled as "outliers" and chose the corresponding quantile. Or you can use Chebychev's inequality to control the upper limit of the probability to label a value, given the model.
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