a question about the formula of computing $e_{i,j,k_{\text{alt}}}$

[Chiyasa]

very excellent work! After reading the paper, I have a question about the formula of $e_{i,j,k_{\text{alt}}}$:
$$e_{i,j,k_{\text{alt}}} = \max_{k} \left| \log_2 \left( \frac{\text{odds}(n_j = k | n_1, \ldots, n_i = k_{\text{alt}}, \ldots, n_{N})}{\text{odds}(n_j = k | n_1, \ldots, n_i = k_{\text{ref}}, \ldots, n_{N})} \right) \right| \quad \text{with} \quad k \in {A, C, G, T}$$
in the formula, it seems like $n_{j}$ could be any one nucleotide of A/C/G/T, but should be the same nucleotide between the numerator and denominator. My question is: Why 1) ${n_{j}}$ not be fixed to $k_{ref}$ on position $j$ 2) ${n_{j}}$ cannot be different nucleotide on the numerator and denominator?

[pedrotomazsilva]

thank you very much @Chiyasa! On the second point, that is right n_j is the same nucleotide on the numerator and denominator.
On the first point, for this metric our rationale was to capture the highest “potential” an alternative (k_alt) nucleotide has to influence the prediction at position j. That means looking at any nucleotide at j and not only its reference.

[Chiyasa]

thank you very much @Chiyasa! On the second point, that is right n_j is the same nucleotide on the numerator and denominator. On the first point, for this metric our rationale was to capture the highest “potential” an alternative (k_alt) nucleotide has to influence the prediction at position j. That means looking at any nucleotide at j and not only its reference.

Another question is that, in this formula, both in the numerator and denominator, in the condition of probability, $n_{j}$ should always be $k_{ref}$ at position $j$, does not change with $k$ in its outcomes?

[pedrotomazsilva]

yes, the odds are conditioned on position n_j being its reference nucleotide, regardless of k.

[Chiyasa]

yes, the odds are conditioned on position n_j being its reference nucleotide, regardless of k.

Thank you for your response! I have no questions now.