As used in Protein structure analysis, Statistical potentials score decoys by comparing their features to experimentally- determined structures, based on the assumption that the observed distributions of particular features reflect energetics: i.e., a common characteristic is assumed to be energetically favourable.

RAPDF (residue-specific all-atom probability discriminatory function)

Essentially a Naive Bayes classifier trained on a set $C$ of native structures (structures observed experimentally and assumed to be correct).

We wish to evaluate $P(C|\{d_{ij}^{ab}\})$, theprobability the structure is a member of the "correct" set $C$, given it contains the distances {$d_{ij}^{ab}$}. We write this probability, using Baye's theorem as

$P(C|\{d_{ij}^{ab}\}) = P(C) \frac{P(\{d_{ij}^{ab}\}|C)}{P(\{d_{ij}^{ab}\})}$

We then make the assumption that $P(\{d_{ij}^{ab}\}|C)$ factorizes as $\prod\limits_{ij}P(d_{ij}^{ab}|C)$ (i.e. the Naive Bayes assumption!).

The score of each decoy (with features $\{d_{ij}^{ab}\}$) is then just the negative log-likelihood

$S(\{d_{ij}^{ab}\}) = -\ln{P(C|\{d_{ij}^{ab}\})} = \sum_{ij} \ln{\left (\frac{P(d^{ij}_{ab}|C)}{P(d^{ij}_{ab})} \right)}$

where $S$ is the score for the decoy, and $d^{ij}_{ab}$ is the distance between atoms $i$ and $j$, of types $a$ and $b$.

See more explanation here

The probabilites are estimated (as in Naive Bayes) as sample frequencies:

$P(d^{ij}_{ab}|C) = \frac{N(d^{ij}_{ab})}{\sum_d (d^{ij}_{ab})}$

where the distances $d^{ij}_{ab}$ have been discretized, and $N$ means number of occurrences of distance $d$ between residues of type $a$ and $b$ over all native configurations in $C$.

The average over all experimental structures is

$P(d^{ij}_{ab}) = \frac{N(d^{ij}_{ab})}{\sum_d (d^{ij}_{ab})}$ over all structures, not just those in $C$.

However, they further approximate this, and assume (I guess as over all structures, there is more randomness..) that this is independent of $a$ and $b$, and that they can estimate this as the average over all $a$ and $b$ and all structures, in $C$ (because we don't have the whole set of possible structures I guess)

$P(d^{ij}_{ab}) \approx P(d) = \frac{\sum_{ab}N(d^{ij}_{ab})}{\sum_d \sum_{ab} (d^{ij}_{ab})}$ over structres in $C$.