Given a continuous probability density function $p(x)$ and some points $x_1, x_2, \ldots x_n$. Can you find a loss function that describes how well represented the points are by $p(x)$.
I would like the function to describe in general how likely that distribution of points was to be generated by $p(x)$. Moreover if I want the loss function to be linear in the $x_i$. Something of the form $f(x_1) + f(x_2) + \cdots + f(x_n)$ where $f$ is arbitrarily complicated.
example here of 8 points sampled from standard normal (mean 0 stdev 1) ... also shown are likelihood values computed for 4-cases ... one is almost perfect ... one is right width but off center ... one is centered but two wide ... the other centered but too narrow
in each case you can see how the almost perfect case has, on average, a larger pdf value at each sampled location