85% of the samples come from an unknown distribution, the rest come from the same distribution with a larger variance.How to recognize them? Assume I have a data, say columns are the samples, and rows are the features, the problem is that around 85% of the samples come from an unknown distribution but the rest come from the same type of distribution but with a larger variance. Is there any method that I could recognize which samples (or columns) comes from the distribution with larger variance? Thank you very much!
 A: I have no idea what features are in this context, but given $n$ sample values $x_1, x_2, \ldots, x_n$, each of which is either a realization of a random variable with density $f_0(x)$ (hypothesis $\mathsf H_0$) or a realization of a random variable with density $f_1(x)$ (hypothesis
$\mathsf H_1$), then the standard maximum likelihood decision rule $$\Lambda(x_i) = \frac{f_1(x_i)}{f_0(x_i)} ~ \begin{array}{c}\mathsf H_1\\\gtrless\\\mathsf H_0\end{array}~ 1$$ applied to each sample will classify the samples into those which are from distribution $f_0(x)$ and those which are from distribution $f_1(x)$. Unfortunately, this rule does not minimize the probability of mis-classification which I assume is one of the goals of the exercise. For the minimum probability of mis-classification, one must compare $\Lambda(x_i)$ to the threshold
$\displaystyle \frac{\pi_0}{\pi_1}$ where $\pi_0$ and $\pi_1$ are the
prior probabilities of the two hypotheses ($0.85$ and $0.15$ in this case), and decide in favor
of $\mathsf H_1$ or $\mathsf H_0$ according as $\Lambda(x(i))$ is larger than or smaller than the threshold $\displaystyle \frac{\pi_0}{\pi_1}$.
For the specific case when $f_0((x)$ and $f_1(x)$ are normal densities with common mean $\mu$ and standard deviations $\sigma_0 < \sigma_1$, the minimum probability of mis-classification rule works out to be what one would expect from common sense: samples that are greatly different from the mean are classified as having density $f_1$ while samples close to the mean are classified as having density $f_0$. The decision rule merely quantifies the exact location of the dividing line(s) between large deviations from the mean and small deviations from the mean. Note that with normal distributions,  the bulk of the probability mass lies in the vicinity of the mean, and consequently, the Type II error probability is quite large.
