# How come Logs-Odds Scoring Matrix is symmetric?

I am studying Pevsner's Bioinformatics.

It is stated that Logs-Odds Scoring Matrix is symmetric at page 89.

But with given the equation to calculate the cells of the matrix, I find different scores for from cysteine to leucine and for from leucine to cysteine.

How is this possible? Do we take the lowest value?

The equation used for constructing Logs-Odds Matrix from a corresponding Pam matrix is : $s_{ij}=10\times\log_{10}(M_{ij}/f_i)$

where $M_{ij}$ is the observed frequency of substitution for each pair aminoacids. and $f_i$ is the probability of aminoacid residue i occurring in the second sequence by chance.

• I'm voting to close this question as off-topic because it is about statistics, not the underlying biology. Please ask at Cross Validated. – MattDMo Nov 19 '16 at 18:16
• To answer this will require more information, given that most respondents will not have p. 89 handy. What is the equation you speak of? – kmm Nov 19 '16 at 22:23

## 1 Answer

Since I do not know what calculations you did. I am going to answer your "How come Logs-Odds Scoring Matrix is symmetric?". Let us assume that it is not. So \begin{equation} S_{ij} \neq S_{ji} \end{equation} \begin{equation} M_{ij}*f_j \neq M_{ji}*f_i \end{equation} Is this not in contradiction with Bayes theorem?. Yes. So log odds matrix is symmetric.