If I have an observation $x$ with a Gaussian distributed observational error of standard deviation $\sigma$ then the sum of likelihoods of that observation having the error free values $x_1^{\prime} \dots x_n^{\prime}$ is
$$p(x\mid x^{\prime}_1\dots x^{\prime}_N, \sigma) = \sum^N_{n=1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp \bigg\{-\frac{(x-x_n^{\prime})^2}{2\sigma^2}\bigg\}$$
A simplified but similar example of my problem is as follows. I have a sample of measured heights of men and women, say $(m_1^{\prime} \dots m_n^{\prime})$ and $(w_1^{\prime} \dots w_n^{\prime})$. I then measure the height $x$ of some individual of unknown gender with error $\sigma$ much larger than the errors on my known samples. Replacing $x^{\prime}$ in the above with $m^{\prime}$ and then $w^{\prime}$ I can compare the two sums of likelihoods and determine whether $x$ is more likely to be a male height or a female height.
But what if instead my samples of men and women heights have comparable errors to $x$. Then my question is how to incorporate this additional error.
Thanks