# Substitution Matrix

#### Suppose that sequences \(x\) and \(y\) are generated based on

### (1) Random model R

#### Letter \(a\) occurs independently with some frequency \(q_a\)

#### 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