# How do I determine how likely some data was drawn from a distribution?

I have a sampled data, say, many vectors $$m_1, \cdots, m_i \in R^{d}$$.

I want an objective function that rates the probability of this data being drawn from certain distribution (here, specifically standard Gaussian $$N(0, \mathbf{I}_{d\times d})$$). That is to say, I want a function $$f(m_1, \cdots, m_i)$$ that takes the vectors $$m_1, \cdots, m_i \in R^{d}$$ and outputs a scalar that, the more likely the vectors are sampled from the distribution $$N(0, \mathbf{I}_{d\times d})$$, the smaller(or bigger) the function's return value is.

Is there any predefined ones in statistics or math theory?

I don't know much about math, just some simple ones from probabilities & statistics undergrad course.

The Mahalanobis distance is a measure of the distance between a point $${\displaystyle P}$$ and a distribution $${\displaystyle D}$$, introduced by P. C. Mahalanobis in 1936... If each of these axes is re-scaled to have unit variance, then the Mahalanobis distance corresponds to standard Euclidean distance in the transformed space. The Mahalanobis distance is thus unitless, scale-invariant, and takes into account the correlations of the data set.
For a normal distribution in any number of dimensions, the probability density of an observation $${\displaystyle {\vec {x}}}$$ is uniquely determined by the Mahalanobis distance $${\displaystyle d}$$... The Mahalanobis distance is proportional, for a normal distribution, to the square root of the negative log-likelihood (after adding a constant so the minimum is at zero).