Substitution Matrix

(1) Random model R

Probability of the two sequence

$P(x,y|R) = \prod_i q_{x_i} \prod_j q_{y_j}$

(2) Match model M

Aligned pair of residues $a$ and $b$ occur with a joint probability $p_{ab}$

Probability of the whole alignment
$P(x,y|M) = \prod_i p_{x_i y_i}$
The ratio of these likelihoods is known as the odds ratio:

For the additive scoring system, take the logarithm (log-odds ratio)
(2.2)
where (2.3)
The scores can be arranged in a matrix,
called as a score matrix or a substitution matrix